Polynomial Regression with Regularization

Clearing and closing the figures
Loading data set
Feature Mapping for Polynomial Regression
Creating training and testing datasets for the learning algorithm
Learning Curves (lambda = 0)
Learning Curves (lambda = 1)
Learning Curves (lambda = 2)
Learning Curves (lambda = 3)
Validation Curve
Training Linear Regression with Regularization (lambda = 0)
Training accuracy of our algorithm using linear regression
Cross Validation Accuracy of our algorithm
Testing our learnt algorithm using linear regression
Training Linear Regression with Regularization (lambda = 1)
Training accuracy of our algorithm using linear regression
Cross Validation Accuracy of our algorithm
Testing our learnt algorithm using linear regression
Training Linear Regression with Regularization (lambda = 2)
Training accuracy of our algorithm using linear regression
Cross Validation Accuracy of our algorithm
Testing our learnt algorithm using linear regression
Training Linear Regression with Regularization (lambda = 2.5)
Training accuracy of our algorithm using linear regression
Cross Validation Accuracy of our algorithm
Testing our learnt algorithm using linear regression

Clearing and closing the figures

close all;
clc;
clear all;

Loading data set

fprintf('Loading dataset...\n\n');
load('comp.mat');
y = comp(:,8);   % assigning target column to variable

Loading dataset...

Feature Mapping for Polynomial Regression

X(:,1) = comp(:,2);   % secon feature is almost linear
X(:,2) = (comp(:,3).^2) / 4;  % third feature is somewhat parabolic
X(:,3) = comp(:,7); % seventh feature is linear
X(:,4) = -1 * log(comp(:,1)) / 0.4342; % first feature is exponential

p = 3; % Polynomial degree to which columns to be raised

% Feature mapping done by poly to higher orders and sinusoids
C = poly(comp(:,[4,5,6]),p);
X = [X C]; % creating a unified feature space
[X, ~, ~] = normalize(X); % normalize the feature data
X = [ones(size(X, 1), 1), X]; % add bias for better learning

Creating training and testing datasets for the learning algorithm

% Training data
Xtrain = X(1:160,:); % disp(size(Xtrain));
ytrain = y(1:160,:); % disp(size(ytrain));

% Cross validation data
Xval = X(161:190,:); % disp(size(xval));
yval = y(161:190,:); % disp(size(yval));

% Testing Data
Xtest = X(191:end,:); % disp(size(Xtest));
ytest = y(191:end,:); % disp(size(ytest));
m = length(ytrain);

Learning Curves (lambda = 0)

% Learning curves help us tackle bias-variance problem. If the training and
% cross validation error converge to an acceptable error level then its
% alright. The following learning curve will help us find the training and
% cross validation performance with different number of training examples

lambda = 0;
[error_train, error_val] = ...
    learningCurve(Xtrain, ytrain, ...
                  Xval, yval, ...
                  lambda);

figure;
plot(1:m, error_train, 1:m, error_val);
title('Learning curve for linear regression (lambda = 0)')
legend('Train', 'Cross Validation')
xlabel('Number of training examples')
ylabel('Error')

Iteration     1 | Cost: 1.405965e-24Iteration     2 | Cost: 4.038968e-28Iteration     3 | Cost: 0.000000e+00
Iteration     1 | Cost: 1.485210e+04Iteration     2 | Cost: 1.690413e+03Iteration     3 | Cost: 1.296485e+03Iteration     4 | Cost: 7.390998e+02Iteration     5 | Cost: 2.225846e+02Iteration     6 | Cost: 1.116555e+02Iteration     7 | Cost: 1.531295e+01Iteration     8 | Cost: 1.789758e+00Iteration     9 | Cost: 1.986645e-03Iteration    10 | Cost: 4.038968e-28Iteration    12 | Cost: 2.019484e-28
Iteration     1 | Cost: 1.716521e+04Iteration     2 | Cost: 3.252070e+03Iteration     3 | Cost: 2.187893e+03Iteration     4 | Cost: 7.009354e+02Iteration     5 | Cost: 7.853260e+01Iteration     6 | Cost: 6.015035e+00Iteration     7 | Cost: 2.236928e+00Iteration     8 | Cost: 2.235353e+00Iteration     9 | Cost: 2.203946e+00Iteration    10 | Cost: 2.162512e+00Iteration    11 | Cost: 1.845452e+00Iteration    12 | Cost: 6.280063e-01Iteration    13 | Cost: 4.425090e-01Iteration    14 | Cost: 4.014189e-01Iteration    15 | Cost: 3.957338e-01Iteration    16 | Cost: 3.774553e-01Iteration    17 | Cost: 3.758514e-01Iteration    18 | Cost: 3.628406e-01Iteration    19 | Cost: 3.497700e-01Iteration    20 | Cost: 2.032624e-01Iteration    21 | Cost: 1.683154e-01Iteration    22 | Cost: 9.162748e-02Iteration    23 | Cost: 5.363357e-03Iteration    24 | Cost: 4.629525e-03Iteration    25 | Cost: 1.873337e-03Iteration    26 | Cost: 5.258955e-04Iteration    27 | Cost: 2.024441e-04Iteration    28 | Cost: 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| Cost: 4.421295e+01Iteration   167 | Cost: 4.420001e+01Iteration   168 | Cost: 4.419772e+01Iteration   169 | Cost: 4.419227e+01Iteration   170 | Cost: 4.418442e+01Iteration   171 | Cost: 4.418086e+01Iteration   172 | Cost: 4.417873e+01Iteration   173 | Cost: 4.417577e+01Iteration   174 | Cost: 4.417105e+01Iteration   175 | Cost: 4.416503e+01Iteration   176 | Cost: 4.416014e+01Iteration   177 | Cost: 4.415407e+01Iteration   178 | Cost: 4.414771e+01Iteration   179 | Cost: 4.414533e+01Iteration   180 | Cost: 4.414317e+01Iteration   181 | Cost: 4.413327e+01Iteration   182 | Cost: 4.412379e+01Iteration   183 | Cost: 4.411558e+01Iteration   184 | Cost: 4.410050e+01Iteration   185 | Cost: 4.406130e+01Iteration   186 | Cost: 4.394844e+01Iteration   187 | Cost: 4.371022e+01Iteration   188 | Cost: 4.337539e+01Iteration   189 | Cost: 4.327578e+01Iteration   190 | Cost: 4.324401e+01Iteration   191 | Cost: 4.322942e+01Iteration   192 | Cost: 4.321435e+01Iteration   193 | Cost: 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Learning Curves (lambda = 1)

% Learning curves help us tackle bias-variance problem. If the training and
% cross validation error converge to an acceptable error level then its
% alright. The following learning curve will help us find the training and
% cross validation performance with different number of training examples

lambda = 1;
[error_train, error_val] = ...
    learningCurve(Xtrain, ytrain, ...
                  Xval, yval, ...
                  lambda);

figure;
plot(1:m, error_train, 1:m, error_val);
title('Learning curve for linear regression (lambda = 1)')
legend('Train', 'Cross Validation')
xlabel('Number of training examples')
ylabel('Error')

Iteration     1 | Cost: 7.402933e+01Iteration     2 | Cost: 5.320275e-25Iteration     3 | Cost: 4.007848e-25Iteration     4 | Cost: 3.860635e-25Iteration     6 | Cost: 3.849832e-25
Iteration     1 | Cost: 1.488669e+04Iteration     2 | Cost: 2.212138e+03Iteration     3 | Cost: 2.098965e+03Iteration     4 | Cost: 1.705818e+03Iteration     5 | Cost: 1.226665e+03Iteration     6 | Cost: 1.150025e+03Iteration     7 | Cost: 1.053371e+03Iteration     8 | Cost: 1.051901e+03Iteration     9 | Cost: 1.047553e+03Iteration    10 | Cost: 1.038669e+03Iteration    11 | Cost: 1.026580e+03Iteration    12 | Cost: 9.919173e+02Iteration    13 | Cost: 7.878375e+02Iteration    14 | Cost: 7.160203e+02Iteration    15 | Cost: 5.650084e+02Iteration    16 | Cost: 3.847243e+02Iteration    17 | Cost: 3.844063e+02Iteration    18 | Cost: 3.810497e+02Iteration    19 | Cost: 3.728505e+02Iteration    20 | Cost: 3.712134e+02Iteration    21 | Cost: 2.327041e+02Iteration    22 | Cost: 1.866936e+02Iteration    23 | Cost: 2.008584e+01Iteration    24 | Cost: 1.698561e+01Iteration    25 | Cost: 8.941540e+00Iteration    26 | Cost: 7.244074e+00Iteration    27 | Cost: 7.129041e+00Iteration    28 | Cost: 7.066227e+00Iteration    29 | Cost: 7.030061e+00Iteration    30 | Cost: 6.567178e+00Iteration    31 | Cost: 6.455260e+00Iteration    32 | Cost: 6.052483e+00Iteration    33 | Cost: 3.417338e+00Iteration    34 | Cost: 2.839361e+00Iteration    35 | Cost: 2.767786e+00Iteration    36 | Cost: 2.701462e+00Iteration    37 | Cost: 2.695702e+00Iteration    38 | Cost: 2.692664e+00Iteration    39 | Cost: 2.689084e+00Iteration    40 | Cost: 2.685209e+00Iteration    41 | Cost: 2.682190e+00Iteration    42 | Cost: 2.678865e+00Iteration    43 | Cost: 2.677242e+00Iteration    44 | Cost: 2.674911e+00Iteration    45 | Cost: 2.674747e+00Iteration    46 | Cost: 2.672751e+00Iteration    47 | Cost: 2.670609e+00Iteration    48 | Cost: 2.667541e+00Iteration    49 | Cost: 2.665764e+00Iteration    50 | Cost: 2.662406e+00Iteration    51 | Cost: 2.653156e+00Iteration    52 | Cost: 2.651539e+00Iteration    53 | Cost: 2.650197e+00Iteration    54 | Cost: 2.649788e+00Iteration    55 | Cost: 2.639491e+00Iteration    56 | Cost: 2.638532e+00Iteration    57 | Cost: 2.597187e+00Iteration    58 | Cost: 2.582286e+00Iteration    59 | Cost: 2.570543e+00Iteration    60 | Cost: 2.569089e+00Iteration    61 | Cost: 2.522328e+00Iteration    62 | Cost: 2.522326e+00Iteration    63 | Cost: 2.522134e+00Iteration    64 | Cost: 2.521616e+00Iteration    65 | Cost: 2.520848e+00Iteration    66 | Cost: 2.520821e+00Iteration    67 | Cost: 2.520808e+00Iteration    68 | Cost: 2.520708e+00Iteration    69 | Cost: 2.520683e+00Iteration    70 | Cost: 2.520682e+00Iteration    71 | Cost: 2.520676e+00Iteration    72 | Cost: 2.520675e+00Iteration    73 | Cost: 2.520669e+00Iteration    74 | Cost: 2.520669e+00Iteration    75 | Cost: 2.520663e+00Iteration    76 | Cost: 2.520657e+00Iteration    77 | Cost: 2.520597e+00Iteration    78 | Cost: 2.520572e+00Iteration    79 | Cost: 2.520567e+00Iteration    80 | Cost: 2.520563e+00Iteration    81 | Cost: 2.520562e+00Iteration    82 | Cost: 2.520562e+00Iteration    83 | Cost: 2.520562e+00Iteration    84 | Cost: 2.520562e+00Iteration    85 | Cost: 2.520562e+00Iteration    86 | Cost: 2.520562e+00Iteration    87 | Cost: 2.520562e+00Iteration    88 | Cost: 2.520562e+00Iteration    89 | Cost: 2.520562e+00Iteration    90 | Cost: 2.520562e+00Iteration    91 | Cost: 2.520562e+00Iteration    92 | Cost: 2.520562e+00Iteration    93 | Cost: 2.520562e+00Iteration    94 | Cost: 2.520562e+00Iteration    96 | Cost: 2.520562e+00
Iteration     1 | Cost: 1.720775e+04Iteration     2 | Cost: 3.532363e+03Iteration     3 | Cost: 2.702506e+03Iteration     4 | Cost: 1.275178e+03Iteration     5 | Cost: 1.215690e+03Iteration     6 | Cost: 7.695190e+02Iteration     7 | Cost: 7.410846e+02Iteration     8 | Cost: 5.542738e+02Iteration     9 | Cost: 4.786611e+02Iteration    10 | Cost: 1.403516e+01Iteration    11 | Cost: 1.231255e+01Iteration    12 | Cost: 7.759646e+00Iteration    13 | Cost: 2.070881e+00Iteration    14 | Cost: 2.059869e+00Iteration    15 | Cost: 1.978815e+00Iteration    16 | Cost: 1.926116e+00Iteration    17 | Cost: 1.853636e+00Iteration    18 | Cost: 1.826170e+00Iteration    19 | Cost: 1.826124e+00Iteration    20 | Cost: 1.789540e+00Iteration    21 | Cost: 1.766504e+00Iteration    22 | Cost: 1.762196e+00Iteration    23 | Cost: 1.715578e+00Iteration    24 | Cost: 1.713856e+00Iteration    25 | Cost: 1.711468e+00Iteration    26 | Cost: 1.711395e+00Iteration    27 | Cost: 1.710441e+00Iteration    28 | Cost: 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| Cost: 1.683955e+00Iteration   167 | Cost: 1.683955e+00Iteration   168 | Cost: 1.683955e+00Iteration   169 | Cost: 1.683955e+00Iteration   170 | Cost: 1.683955e+00Iteration   171 | Cost: 1.683955e+00Iteration   172 | Cost: 1.683955e+00Iteration   174 | Cost: 1.683955e+00Iteration   175 | Cost: 1.683955e+00Iteration   176 | Cost: 1.683955e+00Iteration   177 | Cost: 1.683955e+00Iteration   178 | Cost: 1.683955e+00Iteration   179 | Cost: 1.683955e+00Iteration   180 | Cost: 1.683955e+00Iteration   181 | Cost: 1.683955e+00Iteration   182 | Cost: 1.683955e+00Iteration   183 | Cost: 1.683955e+00Iteration   184 | Cost: 1.683955e+00Iteration   185 | Cost: 1.683955e+00Iteration   186 | Cost: 1.683955e+00Iteration   187 | Cost: 1.683955e+00Iteration   188 | Cost: 1.683955e+00Iteration   189 | Cost: 1.683955e+00Iteration   190 | Cost: 1.683955e+00Iteration   191 | Cost: 1.683955e+00
Iteration     1 | Cost: 1.671112e+04Iteration     2 | Cost: 3.643953e+03Iteration     3 | Cost: 2.174911e+03Iteration     4 | Cost: 5.884827e+02Iteration     5 | Cost: 5.871808e+02Iteration     6 | Cost: 5.814013e+02Iteration     7 | Cost: 5.763121e+02Iteration     8 | Cost: 5.745812e+02Iteration     9 | Cost: 5.719861e+02Iteration    10 | Cost: 5.710017e+02Iteration    11 | Cost: 5.700480e+02Iteration    12 | Cost: 5.618070e+02Iteration    13 | Cost: 5.030953e+02Iteration    14 | Cost: 4.204635e+02Iteration    15 | Cost: 2.576661e+02Iteration    16 | Cost: 1.734360e+02Iteration    17 | Cost: 1.378871e+02Iteration    18 | Cost: 8.922081e+01Iteration    19 | Cost: 8.768500e+01Iteration    20 | Cost: 8.200203e+01Iteration    21 | Cost: 7.809687e+01Iteration    22 | Cost: 7.648332e+01Iteration    23 | Cost: 7.444392e+01Iteration    24 | Cost: 6.945907e+01Iteration    25 | Cost: 6.761745e+01Iteration    26 | Cost: 4.339279e+01Iteration    27 | Cost: 3.620678e+01Iteration    28 | Cost: 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| Cost: 2.318483e+02Iteration    57 | Cost: 2.311951e+02Iteration    58 | Cost: 2.307915e+02Iteration    59 | Cost: 2.303186e+02Iteration    60 | Cost: 2.297833e+02Iteration    61 | Cost: 2.296380e+02Iteration    62 | Cost: 2.291385e+02Iteration    63 | Cost: 2.285722e+02Iteration    64 | Cost: 2.284098e+02Iteration    65 | Cost: 2.283027e+02Iteration    66 | Cost: 2.282359e+02Iteration    67 | Cost: 2.258303e+02Iteration    68 | Cost: 2.245468e+02Iteration    69 | Cost: 2.244176e+02Iteration    70 | Cost: 2.231903e+02Iteration    71 | Cost: 2.230765e+02Iteration    72 | Cost: 2.220911e+02Iteration    73 | Cost: 2.217467e+02Iteration    74 | Cost: 2.207456e+02Iteration    75 | Cost: 2.203386e+02Iteration    76 | Cost: 2.201536e+02Iteration    77 | Cost: 2.197621e+02Iteration    78 | Cost: 2.195884e+02Iteration    79 | Cost: 2.190829e+02Iteration    80 | Cost: 2.186383e+02Iteration    81 | Cost: 2.180621e+02Iteration    82 | Cost: 2.179605e+02Iteration    83 | Cost: 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| Cost: 4.504767e+02Iteration   167 | Cost: 4.504766e+02Iteration   168 | Cost: 4.504765e+02Iteration   169 | Cost: 4.504764e+02Iteration   170 | Cost: 4.504761e+02Iteration   171 | Cost: 4.504759e+02Iteration   172 | Cost: 4.504758e+02Iteration   173 | Cost: 4.504756e+02Iteration   174 | Cost: 4.504756e+02Iteration   175 | Cost: 4.504755e+02Iteration   176 | Cost: 4.504755e+02Iteration   177 | Cost: 4.504755e+02Iteration   178 | Cost: 4.504754e+02Iteration   179 | Cost: 4.504754e+02Iteration   180 | Cost: 4.504754e+02Iteration   181 | Cost: 4.504752e+02Iteration   182 | Cost: 4.504751e+02Iteration   183 | Cost: 4.504746e+02Iteration   184 | Cost: 4.504743e+02Iteration   185 | Cost: 4.504738e+02Iteration   186 | Cost: 4.504737e+02Iteration   187 | Cost: 4.504736e+02Iteration   188 | Cost: 4.504735e+02Iteration   189 | Cost: 4.504734e+02Iteration   190 | Cost: 4.504734e+02Iteration   191 | Cost: 4.504734e+02Iteration   192 | Cost: 4.504733e+02Iteration   193 | Cost: 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| Cost: 4.099040e+02Iteration    57 | Cost: 4.098490e+02Iteration    58 | Cost: 4.098161e+02Iteration    59 | Cost: 4.098146e+02Iteration    60 | Cost: 4.098077e+02Iteration    61 | Cost: 4.098038e+02Iteration    62 | Cost: 4.097970e+02Iteration    63 | Cost: 4.097920e+02Iteration    64 | Cost: 4.097769e+02Iteration    65 | Cost: 4.097729e+02Iteration    66 | Cost: 4.097716e+02Iteration    67 | Cost: 4.097679e+02Iteration    68 | Cost: 4.097659e+02Iteration    69 | Cost: 4.097653e+02Iteration    70 | Cost: 4.097636e+02Iteration    71 | Cost: 4.097626e+02Iteration    72 | Cost: 4.097622e+02Iteration    73 | Cost: 4.097618e+02Iteration    74 | Cost: 4.097614e+02Iteration    75 | Cost: 4.097610e+02Iteration    76 | Cost: 4.097605e+02Iteration    77 | Cost: 4.097601e+02Iteration    78 | Cost: 4.097589e+02Iteration    79 | Cost: 4.097585e+02Iteration    80 | Cost: 4.097582e+02Iteration    81 | Cost: 4.097580e+02Iteration    82 | Cost: 4.097577e+02Iteration    83 | Cost: 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| Cost: 3.648543e+02Iteration    57 | Cost: 3.648511e+02Iteration    58 | Cost: 3.648478e+02Iteration    59 | Cost: 3.648365e+02Iteration    60 | Cost: 3.648297e+02Iteration    61 | Cost: 3.648248e+02Iteration    62 | Cost: 3.648028e+02Iteration    63 | Cost: 3.647964e+02Iteration    64 | Cost: 3.647820e+02Iteration    65 | Cost: 3.647806e+02Iteration    66 | Cost: 3.647775e+02Iteration    67 | Cost: 3.647762e+02Iteration    68 | Cost: 3.647705e+02Iteration    69 | Cost: 3.647703e+02Iteration    70 | Cost: 3.647699e+02Iteration    71 | Cost: 3.647683e+02Iteration    72 | Cost: 3.647678e+02Iteration    73 | Cost: 3.647676e+02Iteration    74 | Cost: 3.647673e+02Iteration    75 | Cost: 3.647663e+02Iteration    76 | Cost: 3.647657e+02Iteration    77 | Cost: 3.647652e+02Iteration    78 | Cost: 3.647650e+02Iteration    79 | Cost: 3.647640e+02Iteration    80 | Cost: 3.647635e+02Iteration    81 | Cost: 3.647633e+02Iteration    82 | Cost: 3.647624e+02Iteration    83 | Cost: 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Iteration     1 | Cost: 5.040417e+03Iteration     2 | Cost: 2.966287e+03Iteration     3 | Cost: 1.227698e+03Iteration     4 | Cost: 7.762874e+02Iteration     5 | Cost: 6.784525e+02Iteration     6 | Cost: 5.669184e+02Iteration     7 | Cost: 5.281076e+02Iteration     8 | Cost: 4.792687e+02Iteration     9 | Cost: 4.620607e+02Iteration    10 | Cost: 4.369318e+02Iteration    11 | Cost: 4.238359e+02Iteration    12 | Cost: 4.171598e+02Iteration    13 | Cost: 3.902736e+02Iteration    14 | Cost: 3.861394e+02Iteration    15 | Cost: 3.834275e+02Iteration    16 | Cost: 3.729670e+02Iteration    17 | Cost: 3.649207e+02Iteration    18 | Cost: 3.577384e+02Iteration    19 | Cost: 3.533367e+02Iteration    20 | Cost: 3.352777e+02Iteration    21 | Cost: 3.351923e+02Iteration    22 | Cost: 3.337763e+02Iteration    23 | Cost: 3.322358e+02Iteration    24 | Cost: 3.304751e+02Iteration    25 | Cost: 3.262768e+02Iteration    26 | Cost: 3.234099e+02Iteration    27 | Cost: 3.222580e+02Iteration    28 | Cost: 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Iteration     1 | Cost: 4.771255e+03Iteration     2 | Cost: 2.884155e+03Iteration     3 | Cost: 1.108677e+03Iteration     4 | Cost: 5.605424e+02Iteration     5 | Cost: 5.270290e+02Iteration     6 | Cost: 4.561211e+02Iteration     7 | Cost: 4.254355e+02Iteration     8 | Cost: 3.602009e+02Iteration     9 | Cost: 3.480217e+02Iteration    10 | Cost: 3.328167e+02Iteration    11 | Cost: 3.284356e+02Iteration    12 | Cost: 3.182317e+02Iteration    13 | Cost: 3.139193e+02Iteration    14 | Cost: 3.105052e+02Iteration    15 | Cost: 3.051303e+02Iteration    16 | Cost: 3.026328e+02Iteration    17 | Cost: 3.019773e+02Iteration    18 | Cost: 2.994872e+02Iteration    19 | Cost: 2.989812e+02Iteration    20 | Cost: 2.976020e+02Iteration    21 | Cost: 2.940397e+02Iteration    22 | Cost: 2.933858e+02Iteration    23 | Cost: 2.913454e+02Iteration    24 | Cost: 2.842519e+02Iteration    25 | Cost: 2.835798e+02Iteration    26 | Cost: 2.827082e+02Iteration    27 | Cost: 2.824661e+02Iteration    28 | Cost: 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| Cost: 2.806941e+02Iteration    57 | Cost: 2.806932e+02Iteration    58 | Cost: 2.806898e+02Iteration    59 | Cost: 2.806887e+02Iteration    60 | Cost: 2.806850e+02Iteration    61 | Cost: 2.806830e+02Iteration    62 | Cost: 2.806824e+02Iteration    63 | Cost: 2.806803e+02Iteration    64 | Cost: 2.806799e+02Iteration    65 | Cost: 2.806793e+02Iteration    66 | Cost: 2.806788e+02Iteration    67 | Cost: 2.806786e+02Iteration    68 | Cost: 2.806785e+02Iteration    69 | Cost: 2.806775e+02Iteration    70 | Cost: 2.806771e+02Iteration    71 | Cost: 2.806764e+02Iteration    72 | Cost: 2.806761e+02Iteration    73 | Cost: 2.806761e+02Iteration    74 | Cost: 2.806759e+02Iteration    75 | Cost: 2.806758e+02Iteration    76 | Cost: 2.806747e+02Iteration    77 | Cost: 2.806730e+02Iteration    78 | Cost: 2.806729e+02Iteration    79 | Cost: 2.806729e+02Iteration    80 | Cost: 2.806728e+02Iteration    81 | Cost: 2.806727e+02Iteration    82 | Cost: 2.806726e+02Iteration    83 | Cost: 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| Cost: 2.806710e+02Iteration   167 | Cost: 2.806710e+02Iteration   168 | Cost: 2.806710e+02Iteration   169 | Cost: 2.806710e+02Iteration   170 | Cost: 2.806710e+02Iteration   171 | Cost: 2.806710e+02Iteration   172 | Cost: 2.806710e+02Iteration   173 | Cost: 2.806710e+02Iteration   174 | Cost: 2.806710e+02Iteration   175 | Cost: 2.806710e+02Iteration   176 | Cost: 2.806710e+02Iteration   177 | Cost: 2.806710e+02Iteration   178 | Cost: 2.806710e+02Iteration   179 | Cost: 2.806710e+02Iteration   180 | Cost: 2.806710e+02Iteration   181 | Cost: 2.806710e+02Iteration   182 | Cost: 2.806710e+02Iteration   183 | Cost: 2.806710e+02Iteration   184 | Cost: 2.806710e+02Iteration   185 | Cost: 2.806710e+02Iteration   186 | Cost: 2.806710e+02Iteration   187 | Cost: 2.806710e+02Iteration   188 | Cost: 2.806710e+02Iteration   189 | Cost: 2.806710e+02Iteration   190 | Cost: 2.806710e+02Iteration   191 | Cost: 2.806710e+02Iteration   192 | Cost: 2.806710e+02Iteration   193 | Cost: 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Iteration     1 | Cost: 4.819061e+03Iteration     2 | Cost: 2.944656e+03Iteration     3 | Cost: 8.009766e+02Iteration     4 | Cost: 6.501046e+02Iteration     5 | Cost: 5.900862e+02Iteration     6 | Cost: 5.057236e+02Iteration     7 | Cost: 4.578947e+02Iteration     8 | Cost: 4.424220e+02Iteration     9 | Cost: 4.214745e+02Iteration    10 | Cost: 4.064325e+02Iteration    11 | Cost: 3.900488e+02Iteration    12 | Cost: 3.787760e+02Iteration    13 | Cost: 3.590528e+02Iteration    14 | Cost: 3.375255e+02Iteration    15 | Cost: 3.290186e+02Iteration    16 | Cost: 3.263906e+02Iteration    17 | Cost: 3.230618e+02Iteration    18 | Cost: 3.193336e+02Iteration    19 | Cost: 3.078683e+02Iteration    20 | Cost: 3.033533e+02Iteration    21 | Cost: 3.003581e+02Iteration    22 | Cost: 2.936682e+02Iteration    23 | Cost: 2.888888e+02Iteration    24 | Cost: 2.862964e+02Iteration    25 | Cost: 2.853086e+02Iteration    26 | Cost: 2.846796e+02Iteration    27 | Cost: 2.795735e+02Iteration    28 | Cost: 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| Cost: 2.688310e+02Iteration    57 | Cost: 2.688301e+02Iteration    58 | Cost: 2.688293e+02Iteration    59 | Cost: 2.688285e+02Iteration    60 | Cost: 2.688259e+02Iteration    61 | Cost: 2.688232e+02Iteration    62 | Cost: 2.688216e+02Iteration    63 | Cost: 2.688203e+02Iteration    64 | Cost: 2.688201e+02Iteration    65 | Cost: 2.688194e+02Iteration    66 | Cost: 2.688193e+02Iteration    67 | Cost: 2.688192e+02Iteration    68 | Cost: 2.688191e+02Iteration    69 | Cost: 2.688190e+02Iteration    70 | Cost: 2.688190e+02Iteration    71 | Cost: 2.688189e+02Iteration    72 | Cost: 2.688189e+02Iteration    73 | Cost: 2.688189e+02Iteration    74 | Cost: 2.688188e+02Iteration    75 | Cost: 2.688188e+02Iteration    76 | Cost: 2.688188e+02Iteration    77 | Cost: 2.688188e+02Iteration    78 | Cost: 2.688188e+02Iteration    79 | Cost: 2.688187e+02Iteration    80 | Cost: 2.688187e+02Iteration    81 | Cost: 2.688187e+02Iteration    82 | Cost: 2.688187e+02Iteration    83 | Cost: 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| Cost: 2.363072e+02Iteration    57 | Cost: 2.363065e+02Iteration    58 | Cost: 2.363006e+02Iteration    59 | Cost: 2.362955e+02Iteration    60 | Cost: 2.362884e+02Iteration    61 | Cost: 2.362823e+02Iteration    62 | Cost: 2.362804e+02Iteration    63 | Cost: 2.362791e+02Iteration    64 | Cost: 2.362737e+02Iteration    65 | Cost: 2.362716e+02Iteration    66 | Cost: 2.362690e+02Iteration    67 | Cost: 2.362645e+02Iteration    68 | Cost: 2.362608e+02Iteration    69 | Cost: 2.362599e+02Iteration    70 | Cost: 2.362575e+02Iteration    71 | Cost: 2.362542e+02Iteration    72 | Cost: 2.362532e+02Iteration    73 | Cost: 2.362526e+02Iteration    74 | Cost: 2.362520e+02Iteration    75 | Cost: 2.362506e+02Iteration    76 | Cost: 2.362497e+02Iteration    77 | Cost: 2.362494e+02Iteration    78 | Cost: 2.362491e+02Iteration    79 | Cost: 2.362489e+02Iteration    80 | Cost: 2.362489e+02Iteration    81 | Cost: 2.362487e+02Iteration    82 | Cost: 2.362485e+02Iteration    83 | Cost: 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| Cost: 9.999077e+01Iteration    57 | Cost: 9.999009e+01Iteration    58 | Cost: 9.998930e+01Iteration    59 | Cost: 9.998848e+01Iteration    60 | Cost: 9.998794e+01Iteration    61 | Cost: 9.998763e+01Iteration    62 | Cost: 9.998753e+01Iteration    63 | Cost: 9.998718e+01Iteration    64 | Cost: 9.998698e+01Iteration    65 | Cost: 9.998687e+01Iteration    66 | Cost: 9.998671e+01Iteration    67 | Cost: 9.998660e+01Iteration    68 | Cost: 9.998651e+01Iteration    69 | Cost: 9.998628e+01Iteration    70 | Cost: 9.998597e+01Iteration    71 | Cost: 9.998568e+01Iteration    72 | Cost: 9.998561e+01Iteration    73 | Cost: 9.998553e+01Iteration    74 | Cost: 9.998532e+01Iteration    75 | Cost: 9.998504e+01Iteration    76 | Cost: 9.998489e+01Iteration    77 | Cost: 9.998486e+01Iteration    78 | Cost: 9.998471e+01Iteration    79 | Cost: 9.998469e+01Iteration    80 | Cost: 9.998467e+01Iteration    81 | Cost: 9.998463e+01Iteration    82 | Cost: 9.998458e+01Iteration    83 | Cost: 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Iteration     1 | Cost: 4.410374e+03Iteration     2 | Cost: 7.947908e+02Iteration     3 | Cost: 3.010774e+02Iteration     4 | Cost: 2.320838e+02Iteration     5 | Cost: 1.895975e+02Iteration     6 | Cost: 1.685346e+02Iteration     7 | Cost: 1.390532e+02Iteration     8 | Cost: 1.267210e+02Iteration     9 | Cost: 1.212578e+02Iteration    10 | Cost: 1.145028e+02Iteration    11 | Cost: 1.121949e+02Iteration    12 | Cost: 1.093794e+02Iteration    13 | Cost: 1.068010e+02Iteration    14 | Cost: 9.962589e+01Iteration    15 | Cost: 9.505614e+01Iteration    16 | Cost: 9.380105e+01Iteration    17 | Cost: 9.314560e+01Iteration    18 | Cost: 9.304920e+01Iteration    19 | Cost: 9.252115e+01Iteration    20 | Cost: 9.207700e+01Iteration    21 | Cost: 9.192907e+01Iteration    22 | Cost: 9.139618e+01Iteration    23 | Cost: 9.105081e+01Iteration    24 | Cost: 9.076049e+01Iteration    25 | Cost: 9.043850e+01Iteration    26 | Cost: 9.031631e+01Iteration    27 | Cost: 9.023555e+01Iteration    28 | Cost: 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Learning Curves (lambda = 2)

% Learning curves help us tackle bias-variance problem. If the training and
% cross validation error converge to an acceptable error level then its
% alright. The following learning curve will help us find the training and
% cross validation performance with different number of training examples

lambda = 0;
[error_train, error_val] = ...
    learningCurve(Xtrain, ytrain, ...
                  Xval, yval, ...
                  lambda);

figure;
plot(1:m, error_train, 1:m, error_val);
title('Learning curve for linear regression (lambda = 2)')
legend('Train', 'Cross Validation')
xlabel('Number of training examples')
ylabel('Error')

Iteration     1 | Cost: 1.405965e-24Iteration     2 | Cost: 4.038968e-28Iteration     3 | Cost: 0.000000e+00
Iteration     1 | Cost: 1.485210e+04Iteration     2 | Cost: 1.690413e+03Iteration     3 | Cost: 1.296485e+03Iteration     4 | Cost: 7.390998e+02Iteration     5 | Cost: 2.225846e+02Iteration     6 | Cost: 1.116555e+02Iteration     7 | Cost: 1.531295e+01Iteration     8 | Cost: 1.789758e+00Iteration     9 | Cost: 1.986645e-03Iteration    10 | Cost: 4.038968e-28Iteration    12 | Cost: 2.019484e-28
Iteration     1 | Cost: 1.716521e+04Iteration     2 | Cost: 3.252070e+03Iteration     3 | Cost: 2.187893e+03Iteration     4 | Cost: 7.009354e+02Iteration     5 | Cost: 7.853260e+01Iteration     6 | Cost: 6.015035e+00Iteration     7 | Cost: 2.236928e+00Iteration     8 | Cost: 2.235353e+00Iteration     9 | Cost: 2.203946e+00Iteration    10 | Cost: 2.162512e+00Iteration    11 | Cost: 1.845452e+00Iteration    12 | Cost: 6.280063e-01Iteration    13 | Cost: 4.425090e-01Iteration    14 | Cost: 4.014189e-01Iteration    15 | Cost: 3.957338e-01Iteration    16 | Cost: 3.774553e-01Iteration    17 | Cost: 3.758514e-01Iteration    18 | Cost: 3.628406e-01Iteration    19 | Cost: 3.497700e-01Iteration    20 | Cost: 2.032624e-01Iteration    21 | Cost: 1.683154e-01Iteration    22 | Cost: 9.162748e-02Iteration    23 | Cost: 5.363357e-03Iteration    24 | Cost: 4.629525e-03Iteration    25 | Cost: 1.873337e-03Iteration    26 | Cost: 5.258955e-04Iteration    27 | Cost: 2.024441e-04Iteration    28 | Cost: 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Iteration     1 | Cost: 4.397760e+03Iteration     2 | Cost: 7.742338e+02Iteration     3 | Cost: 2.803920e+02Iteration     4 | Cost: 2.106263e+02Iteration     5 | Cost: 1.649168e+02Iteration     6 | Cost: 1.397820e+02Iteration     7 | Cost: 1.080338e+02Iteration     8 | Cost: 9.716400e+01Iteration     9 | Cost: 9.021660e+01Iteration    10 | Cost: 8.471952e+01Iteration    11 | Cost: 8.101144e+01Iteration    12 | Cost: 7.809238e+01Iteration    13 | Cost: 6.927287e+01Iteration    14 | Cost: 6.504737e+01Iteration    15 | Cost: 6.340767e+01Iteration    16 | Cost: 6.245677e+01Iteration    17 | Cost: 6.143817e+01Iteration    18 | Cost: 6.005819e+01Iteration    19 | Cost: 5.898202e+01Iteration    20 | Cost: 5.838632e+01Iteration    21 | Cost: 5.766149e+01Iteration    22 | Cost: 5.696689e+01Iteration    23 | Cost: 5.653566e+01Iteration    24 | Cost: 5.630519e+01Iteration    25 | Cost: 5.575686e+01Iteration    26 | Cost: 5.521698e+01Iteration    27 | Cost: 5.486570e+01Iteration    28 | Cost: 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| Cost: 4.535387e+01Iteration   112 | Cost: 4.533839e+01Iteration   113 | Cost: 4.533310e+01Iteration   114 | Cost: 4.533121e+01Iteration   115 | Cost: 4.532315e+01Iteration   116 | Cost: 4.531811e+01Iteration   117 | Cost: 4.531342e+01Iteration   118 | Cost: 4.529025e+01Iteration   119 | Cost: 4.527372e+01Iteration   120 | Cost: 4.526084e+01Iteration   121 | Cost: 4.524667e+01Iteration   122 | Cost: 4.518630e+01Iteration   123 | Cost: 4.516044e+01Iteration   124 | Cost: 4.514675e+01Iteration   125 | Cost: 4.513440e+01Iteration   126 | Cost: 4.508576e+01Iteration   127 | Cost: 4.507738e+01Iteration   128 | Cost: 4.507523e+01Iteration   129 | Cost: 4.507237e+01Iteration   130 | Cost: 4.506501e+01Iteration   131 | Cost: 4.504024e+01Iteration   132 | Cost: 4.501283e+01Iteration   133 | Cost: 4.498662e+01Iteration   134 | Cost: 4.494397e+01Iteration   135 | Cost: 4.491501e+01Iteration   136 | Cost: 4.486273e+01Iteration   137 | Cost: 4.480863e+01Iteration   138 | Cost: 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| Cost: 4.421295e+01Iteration   167 | Cost: 4.420001e+01Iteration   168 | Cost: 4.419772e+01Iteration   169 | Cost: 4.419227e+01Iteration   170 | Cost: 4.418442e+01Iteration   171 | Cost: 4.418086e+01Iteration   172 | Cost: 4.417873e+01Iteration   173 | Cost: 4.417577e+01Iteration   174 | Cost: 4.417105e+01Iteration   175 | Cost: 4.416503e+01Iteration   176 | Cost: 4.416014e+01Iteration   177 | Cost: 4.415407e+01Iteration   178 | Cost: 4.414771e+01Iteration   179 | Cost: 4.414533e+01Iteration   180 | Cost: 4.414317e+01Iteration   181 | Cost: 4.413327e+01Iteration   182 | Cost: 4.412379e+01Iteration   183 | Cost: 4.411558e+01Iteration   184 | Cost: 4.410050e+01Iteration   185 | Cost: 4.406130e+01Iteration   186 | Cost: 4.394844e+01Iteration   187 | Cost: 4.371022e+01Iteration   188 | Cost: 4.337539e+01Iteration   189 | Cost: 4.327578e+01Iteration   190 | Cost: 4.324401e+01Iteration   191 | Cost: 4.322942e+01Iteration   192 | Cost: 4.321435e+01Iteration   193 | Cost: 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Learning Curves (lambda = 3)

% Learning curves help us tackle bias-variance problem. If the training and
% cross validation error converge to an acceptable error level then its
% alright. The following learning curve will help us find the training and
% cross validation performance with different number of training examples

lambda = 0;
[error_train, error_val] = ...
    learningCurve(Xtrain, ytrain, ...
                  Xval, yval, ...
                  lambda);

figure;
plot(1:m, error_train, 1:m, error_val);
title('Learning curve for linear regression (lambda = 3)')
legend('Train', 'Cross Validation')
xlabel('Number of training examples')
ylabel('Error')

Iteration     1 | Cost: 1.405965e-24Iteration     2 | Cost: 4.038968e-28Iteration     3 | Cost: 0.000000e+00
Iteration     1 | Cost: 1.485210e+04Iteration     2 | Cost: 1.690413e+03Iteration     3 | Cost: 1.296485e+03Iteration     4 | Cost: 7.390998e+02Iteration     5 | Cost: 2.225846e+02Iteration     6 | Cost: 1.116555e+02Iteration     7 | Cost: 1.531295e+01Iteration     8 | Cost: 1.789758e+00Iteration     9 | Cost: 1.986645e-03Iteration    10 | Cost: 4.038968e-28Iteration    12 | Cost: 2.019484e-28
Iteration     1 | Cost: 1.716521e+04Iteration     2 | Cost: 3.252070e+03Iteration     3 | Cost: 2.187893e+03Iteration     4 | Cost: 7.009354e+02Iteration     5 | Cost: 7.853260e+01Iteration     6 | Cost: 6.015035e+00Iteration     7 | Cost: 2.236928e+00Iteration     8 | Cost: 2.235353e+00Iteration     9 | Cost: 2.203946e+00Iteration    10 | Cost: 2.162512e+00Iteration    11 | Cost: 1.845452e+00Iteration    12 | Cost: 6.280063e-01Iteration    13 | Cost: 4.425090e-01Iteration    14 | Cost: 4.014189e-01Iteration    15 | Cost: 3.957338e-01Iteration    16 | Cost: 3.774553e-01Iteration    17 | Cost: 3.758514e-01Iteration    18 | Cost: 3.628406e-01Iteration    19 | Cost: 3.497700e-01Iteration    20 | Cost: 2.032624e-01Iteration    21 | Cost: 1.683154e-01Iteration    22 | Cost: 9.162748e-02Iteration    23 | Cost: 5.363357e-03Iteration    24 | Cost: 4.629525e-03Iteration    25 | Cost: 1.873337e-03Iteration    26 | Cost: 5.258955e-04Iteration    27 | Cost: 2.024441e-04Iteration    28 | Cost: 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Iteration     1 | Cost: 4.397760e+03Iteration     2 | Cost: 7.742338e+02Iteration     3 | Cost: 2.803920e+02Iteration     4 | Cost: 2.106263e+02Iteration     5 | Cost: 1.649168e+02Iteration     6 | Cost: 1.397820e+02Iteration     7 | Cost: 1.080338e+02Iteration     8 | Cost: 9.716400e+01Iteration     9 | Cost: 9.021660e+01Iteration    10 | Cost: 8.471952e+01Iteration    11 | Cost: 8.101144e+01Iteration    12 | Cost: 7.809238e+01Iteration    13 | Cost: 6.927287e+01Iteration    14 | Cost: 6.504737e+01Iteration    15 | Cost: 6.340767e+01Iteration    16 | Cost: 6.245677e+01Iteration    17 | Cost: 6.143817e+01Iteration    18 | Cost: 6.005819e+01Iteration    19 | Cost: 5.898202e+01Iteration    20 | Cost: 5.838632e+01Iteration    21 | Cost: 5.766149e+01Iteration    22 | Cost: 5.696689e+01Iteration    23 | Cost: 5.653566e+01Iteration    24 | Cost: 5.630519e+01Iteration    25 | Cost: 5.575686e+01Iteration    26 | Cost: 5.521698e+01Iteration    27 | Cost: 5.486570e+01Iteration    28 | Cost: 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Validation Curve

% Validation curve help us find the optimum lambda parameter which can be
% useful for appreciable regularization. Here we learning curve obtained is
% the training and the testing curve with varying lambda values.

[lambda_vec, error_train, error_val] = ...
    validationCurve(Xtrain, ytrain, Xval, yval);

close all;
figure;
plot(lambda_vec, error_train, lambda_vec, error_val);
legend('Train', 'Cross Validation');
xlabel('lambda');
ylabel('Error');

Iteration     1 | Cost: 1.405965e-24Iteration     2 | Cost: 4.038968e-28Iteration     3 | Cost: 0.000000e+00
Iteration     1 | Cost: 1.485210e+04Iteration     2 | Cost: 1.690413e+03Iteration     3 | Cost: 1.296485e+03Iteration     4 | Cost: 7.390998e+02Iteration     5 | Cost: 2.225846e+02Iteration     6 | Cost: 1.116555e+02Iteration     7 | Cost: 1.531295e+01Iteration     8 | Cost: 1.789758e+00Iteration     9 | Cost: 1.986645e-03Iteration    10 | Cost: 4.038968e-28Iteration    12 | Cost: 2.019484e-28
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| Cost: 1.113019e+02Iteration    57 | Cost: 1.112764e+02Iteration    58 | Cost: 1.112436e+02Iteration    59 | Cost: 1.112074e+02Iteration    60 | Cost: 1.111767e+02Iteration    61 | Cost: 1.110906e+02Iteration    62 | Cost: 1.110568e+02Iteration    63 | Cost: 1.110100e+02Iteration    64 | Cost: 1.109956e+02Iteration    65 | Cost: 1.109427e+02Iteration    66 | Cost: 1.109269e+02Iteration    67 | Cost: 1.109173e+02Iteration    68 | Cost: 1.108741e+02Iteration    69 | Cost: 1.108669e+02Iteration    70 | Cost: 1.108440e+02Iteration    71 | Cost: 1.108282e+02Iteration    72 | Cost: 1.108082e+02Iteration    73 | Cost: 1.107760e+02Iteration    74 | Cost: 1.107466e+02Iteration    75 | Cost: 1.107334e+02Iteration    76 | Cost: 1.107158e+02Iteration    77 | Cost: 1.107136e+02Iteration    78 | Cost: 1.107048e+02Iteration    79 | Cost: 1.106961e+02Iteration    80 | Cost: 1.106798e+02Iteration    81 | Cost: 1.106682e+02Iteration    82 | Cost: 1.106598e+02Iteration    83 | Cost: 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| Cost: 2.442827e+02Iteration   167 | Cost: 2.442824e+02Iteration   168 | Cost: 2.442821e+02Iteration   169 | Cost: 2.442817e+02Iteration   170 | Cost: 2.442815e+02Iteration   171 | Cost: 2.442811e+02Iteration   172 | Cost: 2.442808e+02Iteration   173 | Cost: 2.442807e+02Iteration   174 | Cost: 2.442806e+02Iteration   175 | Cost: 2.442805e+02Iteration   176 | Cost: 2.442804e+02Iteration   177 | Cost: 2.442803e+02Iteration   178 | Cost: 2.442803e+02Iteration   179 | Cost: 2.442802e+02Iteration   180 | Cost: 2.442801e+02Iteration   181 | Cost: 2.442799e+02Iteration   182 | Cost: 2.442795e+02Iteration   183 | Cost: 2.442789e+02Iteration   184 | Cost: 2.442782e+02Iteration   185 | Cost: 2.442780e+02Iteration   186 | Cost: 2.442773e+02Iteration   187 | Cost: 2.442770e+02Iteration   188 | Cost: 2.442769e+02Iteration   189 | Cost: 2.442766e+02Iteration   190 | Cost: 2.442765e+02Iteration   191 | Cost: 2.442764e+02Iteration   192 | Cost: 2.442763e+02Iteration   193 | Cost: 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| Cost: 3.199044e+02Iteration   167 | Cost: 3.199042e+02Iteration   168 | Cost: 3.199037e+02Iteration   169 | Cost: 3.199030e+02Iteration   170 | Cost: 3.199028e+02Iteration   171 | Cost: 3.199026e+02Iteration   172 | Cost: 3.199026e+02Iteration   173 | Cost: 3.199024e+02Iteration   174 | Cost: 3.199023e+02Iteration   175 | Cost: 3.199022e+02Iteration   176 | Cost: 3.199020e+02Iteration   177 | Cost: 3.199018e+02Iteration   178 | Cost: 3.199016e+02Iteration   179 | Cost: 3.199013e+02Iteration   180 | Cost: 3.199013e+02Iteration   181 | Cost: 3.199012e+02Iteration   182 | Cost: 3.199011e+02Iteration   183 | Cost: 3.199011e+02Iteration   184 | Cost: 3.199008e+02Iteration   185 | Cost: 3.199007e+02Iteration   186 | Cost: 3.199006e+02Iteration   187 | Cost: 3.199005e+02Iteration   188 | Cost: 3.199004e+02Iteration   189 | Cost: 3.199003e+02Iteration   190 | Cost: 3.198998e+02Iteration   191 | Cost: 3.198989e+02Iteration   192 | Cost: 3.198979e+02Iteration   193 | Cost: 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| Cost: 4.513446e+02Iteration    57 | Cost: 4.512531e+02Iteration    58 | Cost: 4.512344e+02Iteration    59 | Cost: 4.511820e+02Iteration    60 | Cost: 4.511788e+02Iteration    61 | Cost: 4.511531e+02Iteration    62 | Cost: 4.511153e+02Iteration    63 | Cost: 4.511120e+02Iteration    64 | Cost: 4.511074e+02Iteration    65 | Cost: 4.510968e+02Iteration    66 | Cost: 4.510838e+02Iteration    67 | Cost: 4.510815e+02Iteration    68 | Cost: 4.510743e+02Iteration    69 | Cost: 4.510704e+02Iteration    70 | Cost: 4.510690e+02Iteration    71 | Cost: 4.510625e+02Iteration    72 | Cost: 4.510565e+02Iteration    73 | Cost: 4.510515e+02Iteration    74 | Cost: 4.510459e+02Iteration    75 | Cost: 4.510429e+02Iteration    76 | Cost: 4.510382e+02Iteration    77 | Cost: 4.510103e+02Iteration    78 | Cost: 4.509831e+02Iteration    79 | Cost: 4.509418e+02Iteration    80 | Cost: 4.509325e+02Iteration    81 | Cost: 4.509068e+02Iteration    82 | Cost: 4.509017e+02Iteration    83 | Cost: 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| Cost: 3.956852e+02Iteration    57 | Cost: 3.956620e+02Iteration    58 | Cost: 3.956320e+02Iteration    59 | Cost: 3.956295e+02Iteration    60 | Cost: 3.956187e+02Iteration    61 | Cost: 3.956022e+02Iteration    62 | Cost: 3.955672e+02Iteration    63 | Cost: 3.955221e+02Iteration    64 | Cost: 3.955034e+02Iteration    65 | Cost: 3.954940e+02Iteration    66 | Cost: 3.954846e+02Iteration    67 | Cost: 3.954727e+02Iteration    68 | Cost: 3.953943e+02Iteration    69 | Cost: 3.953070e+02Iteration    70 | Cost: 3.952564e+02Iteration    71 | Cost: 3.952172e+02Iteration    72 | Cost: 3.951880e+02Iteration    73 | Cost: 3.951786e+02Iteration    74 | Cost: 3.951746e+02Iteration    75 | Cost: 3.951709e+02Iteration    76 | Cost: 3.951664e+02Iteration    77 | Cost: 3.951610e+02Iteration    78 | Cost: 3.951560e+02Iteration    79 | Cost: 3.951476e+02Iteration    80 | Cost: 3.951456e+02Iteration    81 | Cost: 3.951371e+02Iteration    82 | Cost: 3.951302e+02Iteration    83 | Cost: 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| Cost: 3.950826e+02Iteration   112 | Cost: 3.950823e+02Iteration   113 | Cost: 3.950821e+02Iteration   114 | Cost: 3.950820e+02Iteration   115 | Cost: 3.950820e+02Iteration   116 | Cost: 3.950818e+02Iteration   117 | Cost: 3.950817e+02Iteration   118 | Cost: 3.950816e+02Iteration   119 | Cost: 3.950816e+02Iteration   120 | Cost: 3.950815e+02Iteration   121 | Cost: 3.950815e+02Iteration   122 | Cost: 3.950815e+02Iteration   123 | Cost: 3.950814e+02Iteration   124 | Cost: 3.950814e+02Iteration   125 | Cost: 3.950814e+02Iteration   126 | Cost: 3.950812e+02Iteration   127 | Cost: 3.950810e+02Iteration   128 | Cost: 3.950810e+02Iteration   129 | Cost: 3.950809e+02Iteration   130 | Cost: 3.950809e+02Iteration   131 | Cost: 3.950809e+02Iteration   132 | Cost: 3.950809e+02Iteration   133 | Cost: 3.950809e+02Iteration   134 | Cost: 3.950809e+02Iteration   135 | Cost: 3.950809e+02Iteration   136 | Cost: 3.950809e+02Iteration   137 | Cost: 3.950809e+02Iteration   138 | Cost: 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| Cost: 3.217369e+02Iteration    57 | Cost: 3.217231e+02Iteration    58 | Cost: 3.217170e+02Iteration    59 | Cost: 3.217159e+02Iteration    60 | Cost: 3.217060e+02Iteration    61 | Cost: 3.217042e+02Iteration    62 | Cost: 3.216990e+02Iteration    63 | Cost: 3.216976e+02Iteration    64 | Cost: 3.216961e+02Iteration    65 | Cost: 3.216942e+02Iteration    66 | Cost: 3.216938e+02Iteration    67 | Cost: 3.216928e+02Iteration    68 | Cost: 3.216917e+02Iteration    69 | Cost: 3.216911e+02Iteration    70 | Cost: 3.216897e+02Iteration    71 | Cost: 3.216893e+02Iteration    72 | Cost: 3.216887e+02Iteration    73 | Cost: 3.216885e+02Iteration    74 | Cost: 3.216885e+02Iteration    75 | Cost: 3.216880e+02Iteration    76 | Cost: 3.216878e+02Iteration    77 | Cost: 3.216875e+02Iteration    78 | Cost: 3.216868e+02Iteration    79 | Cost: 3.216845e+02Iteration    80 | Cost: 3.216838e+02Iteration    81 | Cost: 3.216833e+02Iteration    82 | Cost: 3.216827e+02Iteration    83 | Cost: 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Iteration     1 | Cost: 4.840401e+03Iteration     2 | Cost: 2.914187e+03Iteration     3 | Cost: 1.219051e+03Iteration     4 | Cost: 6.121520e+02Iteration     5 | Cost: 5.596546e+02Iteration     6 | Cost: 4.987848e+02Iteration     7 | Cost: 4.691155e+02Iteration     8 | Cost: 4.277720e+02Iteration     9 | Cost: 4.147551e+02Iteration    10 | Cost: 3.995750e+02Iteration    11 | Cost: 3.803242e+02Iteration    12 | Cost: 3.766189e+02Iteration    13 | Cost: 3.713321e+02Iteration    14 | Cost: 3.669003e+02Iteration    15 | Cost: 3.502564e+02Iteration    16 | Cost: 3.411620e+02Iteration    17 | Cost: 3.366182e+02Iteration    18 | Cost: 3.304923e+02Iteration    19 | Cost: 3.267774e+02Iteration    20 | Cost: 3.246622e+02Iteration    21 | Cost: 3.239425e+02Iteration    22 | Cost: 3.158850e+02Iteration    23 | Cost: 3.139506e+02Iteration    24 | Cost: 3.097686e+02Iteration    25 | Cost: 3.068095e+02Iteration    26 | Cost: 3.048042e+02Iteration    27 | Cost: 3.020045e+02Iteration    28 | Cost: 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| Cost: 2.969545e+02Iteration    57 | Cost: 2.969215e+02Iteration    58 | Cost: 2.969072e+02Iteration    59 | Cost: 2.968860e+02Iteration    60 | Cost: 2.968823e+02Iteration    61 | Cost: 2.968772e+02Iteration    62 | Cost: 2.968753e+02Iteration    63 | Cost: 2.968627e+02Iteration    64 | Cost: 2.968450e+02Iteration    65 | Cost: 2.968412e+02Iteration    66 | Cost: 2.968378e+02Iteration    67 | Cost: 2.968362e+02Iteration    68 | Cost: 2.968345e+02Iteration    69 | Cost: 2.968323e+02Iteration    70 | Cost: 2.968306e+02Iteration    71 | Cost: 2.968255e+02Iteration    72 | Cost: 2.968229e+02Iteration    73 | Cost: 2.968143e+02Iteration    74 | Cost: 2.968102e+02Iteration    75 | Cost: 2.968042e+02Iteration    76 | Cost: 2.967992e+02Iteration    77 | Cost: 2.967967e+02Iteration    78 | Cost: 2.967956e+02Iteration    79 | Cost: 2.967942e+02Iteration    80 | Cost: 2.967926e+02Iteration    81 | Cost: 2.967921e+02Iteration    82 | Cost: 2.967917e+02Iteration    83 | Cost: 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| Cost: 2.721474e+02Iteration    57 | Cost: 2.721460e+02Iteration    58 | Cost: 2.721440e+02Iteration    59 | Cost: 2.721435e+02Iteration    60 | Cost: 2.721418e+02Iteration    61 | Cost: 2.721399e+02Iteration    62 | Cost: 2.721390e+02Iteration    63 | Cost: 2.721366e+02Iteration    64 | Cost: 2.721351e+02Iteration    65 | Cost: 2.721333e+02Iteration    66 | Cost: 2.721320e+02Iteration    67 | Cost: 2.721318e+02Iteration    68 | Cost: 2.721312e+02Iteration    69 | Cost: 2.721303e+02Iteration    70 | Cost: 2.721302e+02Iteration    71 | Cost: 2.721300e+02Iteration    72 | Cost: 2.721293e+02Iteration    73 | Cost: 2.721285e+02Iteration    74 | Cost: 2.721280e+02Iteration    75 | Cost: 2.721270e+02Iteration    76 | Cost: 2.721267e+02Iteration    77 | Cost: 2.721262e+02Iteration    78 | Cost: 2.721261e+02Iteration    79 | Cost: 2.721259e+02Iteration    80 | Cost: 2.721259e+02Iteration    81 | Cost: 2.721259e+02Iteration    82 | Cost: 2.721257e+02Iteration    83 | Cost: 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Iteration     1 | Cost: 8.526581e+03Iteration     2 | Cost: 2.771371e+03Iteration     3 | Cost: 1.188896e+03Iteration     4 | Cost: 8.638571e+02Iteration     5 | Cost: 7.192687e+02Iteration     6 | Cost: 6.990210e+02Iteration     7 | Cost: 6.857673e+02Iteration     8 | Cost: 6.773575e+02Iteration     9 | Cost: 6.699882e+02Iteration    10 | Cost: 6.643860e+02Iteration    11 | Cost: 6.618918e+02Iteration    12 | Cost: 6.595281e+02Iteration    13 | Cost: 6.588011e+02Iteration    14 | Cost: 6.583225e+02Iteration    15 | Cost: 6.581170e+02Iteration    16 | Cost: 6.580417e+02Iteration    17 | Cost: 6.579948e+02Iteration    18 | Cost: 6.579663e+02Iteration    19 | Cost: 6.579404e+02Iteration    20 | Cost: 6.579258e+02Iteration    21 | Cost: 6.579222e+02Iteration    22 | Cost: 6.579206e+02Iteration    23 | Cost: 6.579186e+02Iteration    24 | Cost: 6.579180e+02Iteration    25 | Cost: 6.579177e+02Iteration    26 | Cost: 6.579175e+02Iteration    27 | Cost: 6.579175e+02Iteration    28 | Cost: 6.579173e+02Iteration    29 | Cost: 6.579172e+02Iteration    30 | Cost: 6.579172e+02Iteration    31 | Cost: 6.579172e+02Iteration    32 | Cost: 6.579172e+02Iteration    33 | Cost: 6.579172e+02Iteration    34 | Cost: 6.579172e+02Iteration    35 | Cost: 6.579172e+02Iteration    36 | Cost: 6.579172e+02Iteration    37 | Cost: 6.579172e+02Iteration    38 | Cost: 6.579172e+02Iteration    39 | Cost: 6.579172e+02Iteration    40 | Cost: 6.579172e+02Iteration    41 | Cost: 6.579172e+02Iteration    42 | Cost: 6.579172e+02Iteration    43 | Cost: 6.579172e+02Iteration    44 | Cost: 6.579172e+02Iteration    45 | Cost: 6.579172e+02Iteration    46 | Cost: 6.579172e+02Iteration    47 | Cost: 6.579172e+02Iteration    48 | Cost: 6.579172e+02Iteration    49 | Cost: 6.579172e+02Iteration    50 | Cost: 6.579172e+02Iteration    51 | Cost: 6.579172e+02Iteration    52 | Cost: 6.579172e+02Iteration    53 | Cost: 6.579172e+02Iteration    54 | Cost: 6.579172e+02Iteration    55 | Cost: 6.579172e+02Iteration    56 | Cost: 6.579172e+02Iteration    57 | Cost: 6.579172e+02Iteration    58 | Cost: 6.579172e+02Iteration    59 | Cost: 6.579172e+02Iteration    60 | Cost: 6.579172e+02Iteration    61 | Cost: 6.579172e+02Iteration    62 | Cost: 6.579172e+02Iteration    63 | Cost: 6.579172e+02Iteration    64 | Cost: 6.579172e+02Iteration    65 | Cost: 6.579172e+02
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Iteration     1 | Cost: 5.273321e+03Iteration     2 | Cost: 1.320136e+03Iteration     3 | Cost: 7.404492e+02Iteration     4 | Cost: 5.772688e+02Iteration     5 | Cost: 5.238826e+02Iteration     6 | Cost: 5.053062e+02Iteration     7 | Cost: 4.947626e+02Iteration     8 | Cost: 4.932977e+02Iteration     9 | Cost: 4.857664e+02Iteration    10 | Cost: 4.792298e+02Iteration    11 | Cost: 4.765051e+02Iteration    12 | Cost: 4.739822e+02Iteration    13 | Cost: 4.727680e+02Iteration    14 | Cost: 4.712226e+02Iteration    15 | Cost: 4.704185e+02Iteration    16 | Cost: 4.699472e+02Iteration    17 | Cost: 4.696962e+02Iteration    18 | Cost: 4.695289e+02Iteration    19 | Cost: 4.694451e+02Iteration    20 | Cost: 4.694048e+02Iteration    21 | Cost: 4.693942e+02Iteration    22 | Cost: 4.693814e+02Iteration    23 | Cost: 4.693757e+02Iteration    24 | Cost: 4.693756e+02Iteration    25 | Cost: 4.693707e+02Iteration    26 | Cost: 4.693691e+02Iteration    27 | Cost: 4.693678e+02Iteration    28 | Cost: 4.693668e+02Iteration    29 | Cost: 4.693658e+02Iteration    30 | Cost: 4.693646e+02Iteration    31 | Cost: 4.693642e+02Iteration    32 | Cost: 4.693641e+02Iteration    33 | Cost: 4.693640e+02Iteration    34 | Cost: 4.693640e+02Iteration    35 | Cost: 4.693639e+02Iteration    36 | Cost: 4.693638e+02Iteration    37 | Cost: 4.693638e+02Iteration    38 | Cost: 4.693638e+02Iteration    39 | Cost: 4.693638e+02Iteration    40 | Cost: 4.693638e+02Iteration    41 | Cost: 4.693638e+02Iteration    42 | Cost: 4.693638e+02Iteration    43 | Cost: 4.693638e+02Iteration    44 | Cost: 4.693638e+02Iteration    45 | Cost: 4.693638e+02Iteration    46 | Cost: 4.693638e+02Iteration    47 | Cost: 4.693638e+02Iteration    48 | Cost: 4.693638e+02Iteration    49 | Cost: 4.693638e+02Iteration    50 | Cost: 4.693638e+02Iteration    51 | Cost: 4.693638e+02Iteration    52 | Cost: 4.693638e+02Iteration    53 | Cost: 4.693638e+02Iteration    54 | Cost: 4.693638e+02Iteration    55 | Cost: 4.693638e+02Iteration    56 | Cost: 4.693638e+02Iteration    57 | Cost: 4.693638e+02Iteration    58 | Cost: 4.693638e+02Iteration    59 | Cost: 4.693638e+02Iteration    60 | Cost: 4.693638e+02Iteration    61 | Cost: 4.693638e+02Iteration    62 | Cost: 4.693638e+02Iteration    63 | Cost: 4.693638e+02Iteration    64 | Cost: 4.693638e+02Iteration    65 | Cost: 4.693638e+02Iteration    67 | Cost: 4.693638e+02Iteration    68 | Cost: 4.693638e+02Iteration    69 | Cost: 4.693638e+02Iteration    70 | Cost: 4.693638e+02Iteration    71 | Cost: 4.693638e+02Iteration    72 | Cost: 4.693638e+02Iteration    73 | Cost: 4.693638e+02
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Iteration     1 | Cost: 4.547305e+03Iteration     2 | Cost: 9.763696e+02Iteration     3 | Cost: 4.864433e+02Iteration     4 | Cost: 4.219857e+02Iteration     5 | Cost: 3.851421e+02Iteration     6 | Cost: 3.638584e+02Iteration     7 | Cost: 3.602347e+02Iteration     8 | Cost: 3.491284e+02Iteration     9 | Cost: 3.442995e+02Iteration    10 | Cost: 3.424332e+02Iteration    11 | Cost: 3.414712e+02Iteration    12 | Cost: 3.405527e+02Iteration    13 | Cost: 3.396358e+02Iteration    14 | Cost: 3.393891e+02Iteration    15 | Cost: 3.393233e+02Iteration    16 | Cost: 3.392330e+02Iteration    17 | Cost: 3.392117e+02Iteration    18 | Cost: 3.391818e+02Iteration    19 | Cost: 3.391591e+02Iteration    20 | Cost: 3.391298e+02Iteration    21 | Cost: 3.391235e+02Iteration    22 | Cost: 3.391197e+02Iteration    23 | Cost: 3.391114e+02Iteration    24 | Cost: 3.391052e+02Iteration    25 | Cost: 3.391036e+02Iteration    26 | Cost: 3.391018e+02Iteration    27 | Cost: 3.391008e+02Iteration    28 | Cost: 3.391003e+02Iteration    29 | Cost: 3.390993e+02Iteration    30 | Cost: 3.390991e+02Iteration    31 | Cost: 3.390990e+02Iteration    32 | Cost: 3.390989e+02Iteration    33 | Cost: 3.390988e+02Iteration    34 | Cost: 3.390987e+02Iteration    35 | Cost: 3.390987e+02Iteration    36 | Cost: 3.390986e+02Iteration    37 | Cost: 3.390986e+02Iteration    38 | Cost: 3.390986e+02Iteration    39 | Cost: 3.390985e+02Iteration    40 | Cost: 3.390985e+02Iteration    41 | Cost: 3.390985e+02Iteration    42 | Cost: 3.390985e+02Iteration    43 | Cost: 3.390985e+02Iteration    44 | Cost: 3.390985e+02Iteration    45 | Cost: 3.390985e+02Iteration    46 | Cost: 3.390985e+02Iteration    47 | Cost: 3.390985e+02Iteration    48 | Cost: 3.390985e+02Iteration    49 | Cost: 3.390985e+02Iteration    50 | Cost: 3.390985e+02Iteration    51 | Cost: 3.390985e+02Iteration    52 | Cost: 3.390985e+02Iteration    53 | Cost: 3.390985e+02Iteration    54 | Cost: 3.390985e+02Iteration    55 | Cost: 3.390985e+02Iteration    56 | Cost: 3.390985e+02Iteration    57 | Cost: 3.390985e+02Iteration    58 | Cost: 3.390985e+02Iteration    59 | Cost: 3.390985e+02Iteration    60 | Cost: 3.390985e+02Iteration    61 | Cost: 3.390985e+02Iteration    62 | Cost: 3.390985e+02Iteration    63 | Cost: 3.390985e+02Iteration    64 | Cost: 3.390985e+02Iteration    65 | Cost: 3.390985e+02Iteration    66 | Cost: 3.390985e+02Iteration    67 | Cost: 3.390985e+02Iteration    68 | Cost: 3.390985e+02Iteration    69 | Cost: 3.390985e+02Iteration    70 | Cost: 3.390985e+02Iteration    71 | Cost: 3.390985e+02Iteration    72 | Cost: 3.390985e+02Iteration    73 | Cost: 3.390985e+02Iteration    74 | Cost: 3.390985e+02Iteration    75 | Cost: 3.390985e+02Iteration    76 | Cost: 3.390985e+02Iteration    77 | Cost: 3.390985e+02Iteration    78 | Cost: 3.390985e+02Iteration    79 | Cost: 3.390985e+02Iteration    80 | Cost: 3.390985e+02Iteration    81 | Cost: 3.390985e+02Iteration    83 | Cost: 3.390985e+02
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Iteration     1 | Cost: 4.522393e+03Iteration     2 | Cost: 9.757872e+02Iteration     3 | Cost: 4.814779e+02Iteration     4 | Cost: 4.175267e+02Iteration     5 | Cost: 3.808442e+02Iteration     6 | Cost: 3.593543e+02Iteration     7 | Cost: 3.555245e+02Iteration     8 | Cost: 3.446571e+02Iteration     9 | Cost: 3.400402e+02Iteration    10 | Cost: 3.382325e+02Iteration    11 | Cost: 3.373089e+02Iteration    12 | Cost: 3.363443e+02Iteration    13 | Cost: 3.355169e+02Iteration    14 | Cost: 3.352887e+02Iteration    15 | Cost: 3.351313e+02Iteration    16 | Cost: 3.350804e+02Iteration    17 | Cost: 3.350091e+02Iteration    18 | Cost: 3.349837e+02Iteration    19 | Cost: 3.349680e+02Iteration    20 | Cost: 3.349449e+02Iteration    21 | Cost: 3.349369e+02Iteration    22 | Cost: 3.349302e+02Iteration    23 | Cost: 3.349212e+02Iteration    24 | Cost: 3.349165e+02Iteration    25 | Cost: 3.349129e+02Iteration    26 | Cost: 3.349122e+02Iteration    27 | Cost: 3.349091e+02Iteration    28 | Cost: 3.349084e+02Iteration    29 | Cost: 3.349072e+02Iteration    30 | Cost: 3.349069e+02Iteration    31 | Cost: 3.349068e+02Iteration    32 | Cost: 3.349057e+02Iteration    33 | Cost: 3.349055e+02Iteration    34 | Cost: 3.349045e+02Iteration    35 | Cost: 3.349040e+02Iteration    36 | Cost: 3.349036e+02Iteration    37 | Cost: 3.349031e+02Iteration    38 | Cost: 3.349030e+02Iteration    39 | Cost: 3.349030e+02Iteration    40 | Cost: 3.349030e+02Iteration    41 | Cost: 3.349029e+02Iteration    42 | Cost: 3.349029e+02Iteration    43 | Cost: 3.349029e+02Iteration    44 | Cost: 3.349029e+02Iteration    45 | Cost: 3.349029e+02Iteration    46 | Cost: 3.349029e+02Iteration    47 | Cost: 3.349029e+02Iteration    48 | Cost: 3.349029e+02Iteration    49 | Cost: 3.349029e+02Iteration    50 | Cost: 3.349029e+02Iteration    51 | Cost: 3.349029e+02Iteration    52 | Cost: 3.349029e+02Iteration    53 | Cost: 3.349029e+02Iteration    54 | Cost: 3.349029e+02Iteration    55 | Cost: 3.349029e+02Iteration    56 | Cost: 3.349029e+02Iteration    57 | Cost: 3.349029e+02Iteration    58 | Cost: 3.349029e+02Iteration    59 | Cost: 3.349029e+02Iteration    60 | Cost: 3.349029e+02Iteration    61 | Cost: 3.349029e+02Iteration    62 | Cost: 3.349029e+02Iteration    63 | Cost: 3.349029e+02Iteration    64 | Cost: 3.349029e+02Iteration    65 | Cost: 3.349029e+02Iteration    66 | Cost: 3.349029e+02Iteration    67 | Cost: 3.349029e+02Iteration    68 | Cost: 3.349029e+02Iteration    69 | Cost: 3.349029e+02Iteration    70 | Cost: 3.349029e+02Iteration    71 | Cost: 3.349029e+02Iteration    72 | Cost: 3.349029e+02Iteration    73 | Cost: 3.349029e+02Iteration    74 | Cost: 3.349029e+02Iteration    75 | Cost: 3.349029e+02Iteration    76 | Cost: 3.349029e+02Iteration    77 | Cost: 3.349029e+02Iteration    78 | Cost: 3.349029e+02Iteration    79 | Cost: 3.349029e+02Iteration    80 | Cost: 3.349029e+02Iteration    81 | Cost: 3.349029e+02Iteration    82 | Cost: 3.349029e+02Iteration    83 | Cost: 3.349029e+02Iteration    84 | Cost: 3.349029e+02Iteration    85 | Cost: 3.349029e+02Iteration    86 | Cost: 3.349029e+02Iteration    87 | Cost: 3.349029e+02Iteration    88 | Cost: 3.349029e+02Iteration    90 | Cost: 3.349029e+02Iteration    91 | Cost: 3.349029e+02Iteration    92 | Cost: 3.349029e+02Iteration    93 | Cost: 3.349029e+02Iteration    94 | Cost: 3.349029e+02Iteration    95 | Cost: 3.349029e+02Iteration    96 | Cost: 3.349029e+02Iteration    97 | Cost: 3.349029e+02Iteration    99 | Cost: 3.349029e+02

Training Linear Regression with Regularization (lambda = 0)

lambda = 0;
[theta] = trainLinearReg(Xtrain, ytrain, lambda);

Iteration     1 | Cost: 4.397760e+03Iteration     2 | Cost: 7.742338e+02Iteration     3 | Cost: 2.803920e+02Iteration     4 | Cost: 2.106263e+02Iteration     5 | Cost: 1.649168e+02Iteration     6 | Cost: 1.397820e+02Iteration     7 | Cost: 1.080338e+02Iteration     8 | Cost: 9.716400e+01Iteration     9 | Cost: 9.021660e+01Iteration    10 | Cost: 8.471952e+01Iteration    11 | Cost: 8.101144e+01Iteration    12 | Cost: 7.809238e+01Iteration    13 | Cost: 6.927287e+01Iteration    14 | Cost: 6.504737e+01Iteration    15 | Cost: 6.340767e+01Iteration    16 | Cost: 6.245677e+01Iteration    17 | Cost: 6.143817e+01Iteration    18 | Cost: 6.005819e+01Iteration    19 | Cost: 5.898202e+01Iteration    20 | Cost: 5.838632e+01Iteration    21 | Cost: 5.766149e+01Iteration    22 | Cost: 5.696689e+01Iteration    23 | Cost: 5.653566e+01Iteration    24 | Cost: 5.630519e+01Iteration    25 | Cost: 5.575686e+01Iteration    26 | Cost: 5.521698e+01Iteration    27 | Cost: 5.486570e+01Iteration    28 | Cost: 5.464352e+01Iteration    29 | Cost: 5.398915e+01Iteration    30 | Cost: 5.263771e+01Iteration    31 | Cost: 5.192705e+01Iteration    32 | Cost: 5.100488e+01Iteration    33 | Cost: 5.075967e+01Iteration    34 | Cost: 5.063748e+01Iteration    35 | Cost: 5.052995e+01Iteration    36 | Cost: 5.034994e+01Iteration    37 | Cost: 5.025294e+01Iteration    38 | Cost: 5.018839e+01Iteration    39 | Cost: 5.009322e+01Iteration    40 | Cost: 5.005372e+01Iteration    41 | Cost: 5.002983e+01Iteration    42 | Cost: 4.979676e+01Iteration    43 | Cost: 4.930055e+01Iteration    44 | Cost: 4.856202e+01Iteration    45 | Cost: 4.823236e+01Iteration    46 | Cost: 4.807656e+01Iteration    47 | Cost: 4.788211e+01Iteration    48 | Cost: 4.786218e+01Iteration    49 | Cost: 4.784914e+01Iteration    50 | Cost: 4.780492e+01Iteration    51 | Cost: 4.768862e+01Iteration    52 | Cost: 4.752466e+01Iteration    53 | Cost: 4.739118e+01Iteration    54 | Cost: 4.732280e+01Iteration    55 | Cost: 4.729429e+01Iteration    56 | Cost: 4.722379e+01Iteration    57 | Cost: 4.714338e+01Iteration    58 | Cost: 4.705278e+01Iteration    59 | Cost: 4.698245e+01Iteration    60 | Cost: 4.688754e+01Iteration    61 | Cost: 4.681797e+01Iteration    62 | Cost: 4.677893e+01Iteration    63 | Cost: 4.674677e+01Iteration    64 | Cost: 4.665809e+01Iteration    65 | Cost: 4.660673e+01Iteration    66 | Cost: 4.657524e+01Iteration    67 | Cost: 4.654958e+01Iteration    68 | Cost: 4.653417e+01Iteration    69 | Cost: 4.649511e+01Iteration    70 | Cost: 4.645380e+01Iteration    71 | Cost: 4.642875e+01Iteration    72 | Cost: 4.640661e+01Iteration    73 | Cost: 4.634244e+01Iteration    74 | Cost: 4.631700e+01Iteration    75 | Cost: 4.629631e+01Iteration    76 | Cost: 4.628341e+01Iteration    77 | Cost: 4.625081e+01Iteration    78 | Cost: 4.621608e+01Iteration    79 | Cost: 4.617478e+01Iteration    80 | Cost: 4.614953e+01Iteration    81 | Cost: 4.613665e+01Iteration    82 | Cost: 4.608387e+01Iteration    83 | Cost: 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| Cost: 4.535387e+01Iteration   112 | Cost: 4.533839e+01Iteration   113 | Cost: 4.533310e+01Iteration   114 | Cost: 4.533121e+01Iteration   115 | Cost: 4.532315e+01Iteration   116 | Cost: 4.531811e+01Iteration   117 | Cost: 4.531342e+01Iteration   118 | Cost: 4.529025e+01Iteration   119 | Cost: 4.527372e+01Iteration   120 | Cost: 4.526084e+01Iteration   121 | Cost: 4.524667e+01Iteration   122 | Cost: 4.518630e+01Iteration   123 | Cost: 4.516044e+01Iteration   124 | Cost: 4.514675e+01Iteration   125 | Cost: 4.513440e+01Iteration   126 | Cost: 4.508576e+01Iteration   127 | Cost: 4.507738e+01Iteration   128 | Cost: 4.507523e+01Iteration   129 | Cost: 4.507237e+01Iteration   130 | Cost: 4.506501e+01Iteration   131 | Cost: 4.504024e+01Iteration   132 | Cost: 4.501283e+01Iteration   133 | Cost: 4.498662e+01Iteration   134 | Cost: 4.494397e+01Iteration   135 | Cost: 4.491501e+01Iteration   136 | Cost: 4.486273e+01Iteration   137 | Cost: 4.480863e+01Iteration   138 | Cost: 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| Cost: 4.421295e+01Iteration   167 | Cost: 4.420001e+01Iteration   168 | Cost: 4.419772e+01Iteration   169 | Cost: 4.419227e+01Iteration   170 | Cost: 4.418442e+01Iteration   171 | Cost: 4.418086e+01Iteration   172 | Cost: 4.417873e+01Iteration   173 | Cost: 4.417577e+01Iteration   174 | Cost: 4.417105e+01Iteration   175 | Cost: 4.416503e+01Iteration   176 | Cost: 4.416014e+01Iteration   177 | Cost: 4.415407e+01Iteration   178 | Cost: 4.414771e+01Iteration   179 | Cost: 4.414533e+01Iteration   180 | Cost: 4.414317e+01Iteration   181 | Cost: 4.413327e+01Iteration   182 | Cost: 4.412379e+01Iteration   183 | Cost: 4.411558e+01Iteration   184 | Cost: 4.410050e+01Iteration   185 | Cost: 4.406130e+01Iteration   186 | Cost: 4.394844e+01Iteration   187 | Cost: 4.371022e+01Iteration   188 | Cost: 4.337539e+01Iteration   189 | Cost: 4.327578e+01Iteration   190 | Cost: 4.324401e+01Iteration   191 | Cost: 4.322942e+01Iteration   192 | Cost: 4.321435e+01Iteration   193 | Cost: 4.320182e+01Iteration   194 | Cost: 4.318830e+01Iteration   195 | Cost: 4.318300e+01Iteration   196 | Cost: 4.318041e+01Iteration   197 | Cost: 4.317670e+01Iteration   198 | Cost: 4.317399e+01Iteration   199 | Cost: 4.317134e+01Iteration   200 | Cost: 4.316089e+01

Training accuracy of our algorithm using linear regression

fprintf('lambda = 0\n');
% Applying learnt parameters on test data
pricetr = Xtrain * theta;

% Showing algorithm accuracy
errortr = abs(pricetr - ytrain) ./ ytrain;
accuracytr = 100 - (mean(errortr) * 100);
fprintf('Training accuracy on training set: %f\n', accuracytr);

lambda = 0
Training accuracy on training set: 88.397794

Cross Validation Accuracy of our algorithm

%disp(size(Xtest));
pricecv = Xval * theta;

% Showing algorithm accuracy
errorcv = abs(pricecv - yval) ./ yval;
accuracycv = 100 - (mean(errorcv) * 100);
fprintf('Cross Validation accuracy on cross validation set: %f\n', accuracycv);

Cross Validation accuracy on cross validation set: 89.027884

Testing our learnt algorithm using linear regression

% Applying learnt parameters on test data
%disp(size(Xtest));
pricete = Xtest * theta;

% Showing algorithm accuracy
errorte = abs(pricete - ytest) ./ ytest;
accuracyte = 100 - (mean(errorte) * 100);
fprintf('Testing accuracy on test set: %f\n\n', accuracyte);

Testing accuracy on test set: 54.542140

Training Linear Regression with Regularization (lambda = 1)

lambda = 1;
[theta1] = trainLinearReg(Xtrain, ytrain, lambda);

Iteration     1 | Cost: 4.410374e+03Iteration     2 | Cost: 7.947908e+02Iteration     3 | Cost: 3.010774e+02Iteration     4 | Cost: 2.320838e+02Iteration     5 | Cost: 1.895975e+02Iteration     6 | Cost: 1.685346e+02Iteration     7 | Cost: 1.390532e+02Iteration     8 | Cost: 1.267210e+02Iteration     9 | Cost: 1.212578e+02Iteration    10 | Cost: 1.145028e+02Iteration    11 | Cost: 1.121949e+02Iteration    12 | Cost: 1.093794e+02Iteration    13 | Cost: 1.068010e+02Iteration    14 | Cost: 9.962589e+01Iteration    15 | Cost: 9.505614e+01Iteration    16 | Cost: 9.380105e+01Iteration    17 | Cost: 9.314560e+01Iteration    18 | Cost: 9.304920e+01Iteration    19 | Cost: 9.252115e+01Iteration    20 | Cost: 9.207700e+01Iteration    21 | Cost: 9.192907e+01Iteration    22 | Cost: 9.139618e+01Iteration    23 | Cost: 9.105081e+01Iteration    24 | Cost: 9.076049e+01Iteration    25 | Cost: 9.043850e+01Iteration    26 | Cost: 9.031631e+01Iteration    27 | Cost: 9.023555e+01Iteration    28 | Cost: 9.013367e+01Iteration    29 | Cost: 8.964195e+01Iteration    30 | Cost: 8.953444e+01Iteration    31 | Cost: 8.946581e+01Iteration    32 | Cost: 8.938600e+01Iteration    33 | Cost: 8.916030e+01Iteration    34 | Cost: 8.882800e+01Iteration    35 | Cost: 8.874548e+01Iteration    36 | Cost: 8.871386e+01Iteration    37 | Cost: 8.869269e+01Iteration    38 | Cost: 8.865333e+01Iteration    39 | Cost: 8.860804e+01Iteration    40 | Cost: 8.856324e+01Iteration    41 | Cost: 8.852470e+01Iteration    42 | Cost: 8.849901e+01Iteration    43 | Cost: 8.846623e+01Iteration    44 | Cost: 8.839515e+01Iteration    45 | Cost: 8.829251e+01Iteration    46 | Cost: 8.825562e+01Iteration    47 | Cost: 8.822310e+01Iteration    48 | Cost: 8.819057e+01Iteration    49 | Cost: 8.816737e+01Iteration    50 | Cost: 8.812612e+01Iteration    51 | Cost: 8.807676e+01Iteration    52 | Cost: 8.804794e+01Iteration    53 | Cost: 8.803213e+01Iteration    54 | Cost: 8.802305e+01Iteration    55 | Cost: 8.801728e+01Iteration    56 | Cost: 8.800433e+01Iteration    57 | Cost: 8.795491e+01Iteration    58 | Cost: 8.791087e+01Iteration    59 | Cost: 8.786639e+01Iteration    60 | Cost: 8.781468e+01Iteration    61 | Cost: 8.780769e+01Iteration    62 | Cost: 8.780414e+01Iteration    63 | Cost: 8.780410e+01Iteration    64 | Cost: 8.780198e+01Iteration    65 | Cost: 8.780073e+01Iteration    66 | Cost: 8.780047e+01Iteration    67 | Cost: 8.779930e+01Iteration    68 | Cost: 8.779859e+01Iteration    69 | Cost: 8.779774e+01Iteration    70 | Cost: 8.779583e+01Iteration    71 | Cost: 8.779382e+01Iteration    72 | Cost: 8.778599e+01Iteration    73 | Cost: 8.778368e+01Iteration    74 | Cost: 8.778107e+01Iteration    75 | Cost: 8.777956e+01Iteration    76 | Cost: 8.777866e+01Iteration    77 | Cost: 8.777615e+01Iteration    78 | Cost: 8.777521e+01Iteration    79 | Cost: 8.777464e+01Iteration    80 | Cost: 8.777382e+01Iteration    81 | Cost: 8.777340e+01Iteration    82 | Cost: 8.777314e+01Iteration    83 | Cost: 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| Cost: 8.775815e+01Iteration   112 | Cost: 8.775813e+01Iteration   113 | Cost: 8.775813e+01Iteration   114 | Cost: 8.775813e+01Iteration   115 | Cost: 8.775811e+01Iteration   116 | Cost: 8.775810e+01Iteration   117 | Cost: 8.775809e+01Iteration   118 | Cost: 8.775807e+01Iteration   119 | Cost: 8.775804e+01Iteration   120 | Cost: 8.775804e+01Iteration   121 | Cost: 8.775803e+01Iteration   122 | Cost: 8.775803e+01Iteration   123 | Cost: 8.775802e+01Iteration   124 | Cost: 8.775800e+01Iteration   125 | Cost: 8.775798e+01Iteration   126 | Cost: 8.775796e+01Iteration   127 | Cost: 8.775795e+01Iteration   128 | Cost: 8.775795e+01Iteration   129 | Cost: 8.775793e+01Iteration   130 | Cost: 8.775793e+01Iteration   131 | Cost: 8.775792e+01Iteration   132 | Cost: 8.775792e+01Iteration   133 | Cost: 8.775792e+01Iteration   134 | Cost: 8.775792e+01Iteration   135 | Cost: 8.775792e+01Iteration   136 | Cost: 8.775792e+01Iteration   137 | Cost: 8.775792e+01Iteration   138 | Cost: 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| Cost: 8.775787e+01Iteration   167 | Cost: 8.775787e+01Iteration   168 | Cost: 8.775787e+01Iteration   169 | Cost: 8.775787e+01Iteration   170 | Cost: 8.775787e+01Iteration   171 | Cost: 8.775787e+01Iteration   172 | Cost: 8.775787e+01Iteration   173 | Cost: 8.775787e+01Iteration   174 | Cost: 8.775787e+01Iteration   175 | Cost: 8.775787e+01Iteration   176 | Cost: 8.775787e+01Iteration   177 | Cost: 8.775787e+01Iteration   178 | Cost: 8.775787e+01Iteration   179 | Cost: 8.775787e+01Iteration   180 | Cost: 8.775787e+01Iteration   181 | Cost: 8.775787e+01Iteration   182 | Cost: 8.775787e+01Iteration   183 | Cost: 8.775787e+01Iteration   184 | Cost: 8.775787e+01Iteration   185 | Cost: 8.775787e+01Iteration   186 | Cost: 8.775787e+01Iteration   187 | Cost: 8.775787e+01Iteration   188 | Cost: 8.775787e+01Iteration   189 | Cost: 8.775787e+01Iteration   190 | Cost: 8.775787e+01Iteration   191 | Cost: 8.775787e+01Iteration   192 | Cost: 8.775787e+01Iteration   193 | Cost: 8.775787e+01Iteration   194 | Cost: 8.775787e+01Iteration   195 | Cost: 8.775787e+01Iteration   196 | Cost: 8.775787e+01Iteration   197 | Cost: 8.775787e+01Iteration   198 | Cost: 8.775787e+01Iteration   199 | Cost: 8.775787e+01Iteration   200 | Cost: 8.775787e+01

Training accuracy of our algorithm using linear regression

fprintf('lambda = 1\n');
% Applying learnt parameters on test data
pricetr = Xtrain * theta1;

% Showing algorithm accuracy
errortr = abs(pricetr - ytrain) ./ ytrain;
accuracytr = 100 - (mean(errortr) * 100);
fprintf('Training accuracy on training set: %f\n', accuracytr);

lambda = 1
Training accuracy on training set: 88.499990

Cross Validation Accuracy of our algorithm

%disp(size(Xtest));
pricecv = Xval * theta1;

% Showing algorithm accuracy
errorcv = abs(pricecv - yval) ./ yval;
accuracycv = 100 - (mean(errorcv) * 100);
fprintf('Cross Validation accuracy on cross validation set: %f\n', accuracycv);

Cross Validation accuracy on cross validation set: 89.631917

Testing our learnt algorithm using linear regression

% Applying learnt parameters on test data
%disp(size(Xtest));
pricete = Xtest * theta1;

% Showing algorithm accuracy
errorte = abs(pricete - ytest) ./ ytest;
accuracyte = 100 - (mean(errorte) * 100);
fprintf('Testing accuracy on test set: %f\n\n', accuracyte);

Testing accuracy on test set: 67.146308

Training Linear Regression with Regularization (lambda = 2)

lambda = 2;
[theta2] = trainLinearReg(Xtrain, ytrain, lambda);

Iteration     1 | Cost: 4.422954e+03Iteration     2 | Cost: 8.152567e+02Iteration     3 | Cost: 3.216306e+02Iteration     4 | Cost: 2.533618e+02Iteration     5 | Cost: 2.140285e+02Iteration     6 | Cost: 1.967307e+02Iteration     7 | Cost: 1.604805e+02Iteration     8 | Cost: 1.506274e+02Iteration     9 | Cost: 1.457408e+02Iteration    10 | Cost: 1.410173e+02Iteration    11 | Cost: 1.357121e+02Iteration    12 | Cost: 1.326819e+02Iteration    13 | Cost: 1.312967e+02Iteration    14 | Cost: 1.277645e+02Iteration    15 | Cost: 1.267623e+02Iteration    16 | Cost: 1.262323e+02Iteration    17 | Cost: 1.259732e+02Iteration    18 | Cost: 1.250981e+02Iteration    19 | Cost: 1.248058e+02Iteration    20 | Cost: 1.245985e+02Iteration    21 | Cost: 1.244228e+02Iteration    22 | Cost: 1.240015e+02Iteration    23 | Cost: 1.236375e+02Iteration    24 | Cost: 1.234188e+02Iteration    25 | Cost: 1.233764e+02Iteration    26 | Cost: 1.232661e+02Iteration    27 | Cost: 1.231284e+02Iteration    28 | Cost: 1.230683e+02Iteration    29 | Cost: 1.229315e+02Iteration    30 | Cost: 1.224051e+02Iteration    31 | Cost: 1.222486e+02Iteration    32 | Cost: 1.221905e+02Iteration    33 | Cost: 1.221298e+02Iteration    34 | Cost: 1.221032e+02Iteration    35 | Cost: 1.220680e+02Iteration    36 | Cost: 1.220421e+02Iteration    37 | Cost: 1.220249e+02Iteration    38 | Cost: 1.219733e+02Iteration    39 | Cost: 1.218969e+02Iteration    40 | Cost: 1.218799e+02Iteration    41 | Cost: 1.218704e+02Iteration    42 | Cost: 1.218641e+02Iteration    43 | Cost: 1.218487e+02Iteration    44 | Cost: 1.218413e+02Iteration    45 | Cost: 1.218341e+02Iteration    46 | Cost: 1.218240e+02Iteration    47 | Cost: 1.218133e+02Iteration    48 | Cost: 1.218072e+02Iteration    49 | Cost: 1.218035e+02Iteration    50 | Cost: 1.217943e+02Iteration    51 | Cost: 1.217830e+02Iteration    52 | Cost: 1.217797e+02Iteration    53 | Cost: 1.217764e+02Iteration    54 | Cost: 1.217720e+02Iteration    55 | Cost: 1.217691e+02Iteration    56 | Cost: 1.217657e+02Iteration    57 | Cost: 1.217583e+02Iteration    58 | Cost: 1.217517e+02Iteration    59 | Cost: 1.217449e+02Iteration    60 | Cost: 1.217378e+02Iteration    61 | Cost: 1.217341e+02Iteration    62 | Cost: 1.217329e+02Iteration    63 | Cost: 1.217285e+02Iteration    64 | Cost: 1.217281e+02Iteration    65 | Cost: 1.217279e+02Iteration    66 | Cost: 1.217277e+02Iteration    67 | Cost: 1.217275e+02Iteration    68 | Cost: 1.217271e+02Iteration    69 | Cost: 1.217265e+02Iteration    70 | Cost: 1.217262e+02Iteration    71 | Cost: 1.217262e+02Iteration    72 | Cost: 1.217260e+02Iteration    73 | Cost: 1.217260e+02Iteration    74 | Cost: 1.217259e+02Iteration    75 | Cost: 1.217258e+02Iteration    76 | Cost: 1.217257e+02Iteration    77 | Cost: 1.217256e+02Iteration    78 | Cost: 1.217255e+02Iteration    79 | Cost: 1.217254e+02Iteration    80 | Cost: 1.217253e+02Iteration    81 | Cost: 1.217253e+02Iteration    82 | Cost: 1.217252e+02Iteration    83 | Cost: 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| Cost: 1.217246e+02Iteration   112 | Cost: 1.217246e+02Iteration   113 | Cost: 1.217246e+02Iteration   114 | Cost: 1.217246e+02Iteration   115 | Cost: 1.217246e+02Iteration   116 | Cost: 1.217246e+02Iteration   117 | Cost: 1.217246e+02Iteration   118 | Cost: 1.217246e+02Iteration   119 | Cost: 1.217246e+02Iteration   120 | Cost: 1.217246e+02Iteration   121 | Cost: 1.217246e+02Iteration   122 | Cost: 1.217246e+02Iteration   123 | Cost: 1.217246e+02Iteration   124 | Cost: 1.217246e+02Iteration   125 | Cost: 1.217246e+02Iteration   126 | Cost: 1.217246e+02Iteration   127 | Cost: 1.217246e+02Iteration   128 | Cost: 1.217246e+02Iteration   129 | Cost: 1.217246e+02Iteration   130 | Cost: 1.217246e+02Iteration   131 | Cost: 1.217246e+02Iteration   132 | Cost: 1.217246e+02Iteration   133 | Cost: 1.217246e+02Iteration   134 | Cost: 1.217246e+02Iteration   135 | Cost: 1.217246e+02Iteration   136 | Cost: 1.217246e+02Iteration   137 | Cost: 1.217246e+02Iteration   138 | Cost: 1.217246e+02Iteration   139 | Cost: 1.217246e+02Iteration   140 | Cost: 1.217246e+02Iteration   141 | Cost: 1.217246e+02Iteration   142 | Cost: 1.217246e+02Iteration   143 | Cost: 1.217246e+02Iteration   144 | Cost: 1.217246e+02Iteration   145 | Cost: 1.217246e+02Iteration   146 | Cost: 1.217246e+02Iteration   147 | Cost: 1.217246e+02Iteration   148 | Cost: 1.217246e+02Iteration   149 | Cost: 1.217246e+02Iteration   150 | Cost: 1.217246e+02Iteration   151 | Cost: 1.217246e+02Iteration   152 | Cost: 1.217246e+02Iteration   153 | Cost: 1.217246e+02Iteration   154 | Cost: 1.217246e+02Iteration   155 | Cost: 1.217246e+02Iteration   156 | Cost: 1.217246e+02Iteration   157 | Cost: 1.217246e+02Iteration   158 | Cost: 1.217246e+02Iteration   159 | Cost: 1.217246e+02Iteration   160 | Cost: 1.217246e+02Iteration   161 | Cost: 1.217246e+02Iteration   162 | Cost: 1.217246e+02Iteration   163 | Cost: 1.217246e+02Iteration   164 | Cost: 1.217246e+02Iteration   165 | Cost: 1.217246e+02Iteration   166 | Cost: 1.217246e+02Iteration   167 | Cost: 1.217246e+02Iteration   168 | Cost: 1.217246e+02Iteration   169 | Cost: 1.217246e+02Iteration   170 | Cost: 1.217246e+02Iteration   171 | Cost: 1.217246e+02Iteration   172 | Cost: 1.217246e+02Iteration   173 | Cost: 1.217246e+02Iteration   174 | Cost: 1.217246e+02Iteration   175 | Cost: 1.217246e+02Iteration   176 | Cost: 1.217246e+02Iteration   177 | Cost: 1.217246e+02Iteration   178 | Cost: 1.217246e+02Iteration   179 | Cost: 1.217246e+02Iteration   180 | Cost: 1.217246e+02Iteration   181 | Cost: 1.217246e+02Iteration   182 | Cost: 1.217246e+02Iteration   183 | Cost: 1.217246e+02Iteration   184 | Cost: 1.217246e+02Iteration   185 | Cost: 1.217246e+02Iteration   186 | Cost: 1.217246e+02Iteration   187 | Cost: 1.217246e+02Iteration   188 | Cost: 1.217246e+02Iteration   189 | Cost: 1.217246e+02Iteration   190 | Cost: 1.217246e+02Iteration   191 | Cost: 1.217246e+02Iteration   192 | Cost: 1.217246e+02Iteration   193 | Cost: 1.217246e+02Iteration   194 | Cost: 1.217246e+02Iteration   195 | Cost: 1.217246e+02Iteration   196 | Cost: 1.217246e+02Iteration   197 | Cost: 1.217246e+02Iteration   198 | Cost: 1.217246e+02Iteration   199 | Cost: 1.217246e+02Iteration   200 | Cost: 1.217246e+02

Training accuracy of our algorithm using linear regression

fprintf('lambda = 2\n');
% Applying learnt parameters on test data
pricetr = Xtrain * theta2;

% Showing algorithm accuracy
errortr = abs(pricetr - ytrain) ./ ytrain;
accuracytr = 100 - (mean(errortr) * 100);
fprintf('Training accuracy on training set: %f\n', accuracytr);

lambda = 2
Training accuracy on training set: 88.018023

Cross Validation Accuracy of our algorithm

%disp(size(Xtest));
pricecv = Xval * theta2;

% Showing algorithm accuracy
errorcv = abs(pricecv - yval) ./ yval;
accuracycv = 100 - (mean(errorcv) * 100);
fprintf('Cross Validation accuracy on cross validation set: %f\n', accuracycv);

Cross Validation accuracy on cross validation set: 89.014401

Testing our learnt algorithm using linear regression

% Applying learnt parameters on test data
%disp(size(Xtest));
pricete = Xtest * theta2;

% Showing algorithm accuracy
errorte = abs(pricete - ytest) ./ ytest;
accuracyte = 100 - (mean(errorte) * 100);
fprintf('Testing accuracy on test set: %f\n\n', accuracyte);

Testing accuracy on test set: 70.205441

Training Linear Regression with Regularization (lambda = 2.5)

lambda = 2.5;
[theta3] = trainLinearReg(Xtrain, ytrain, lambda);

Iteration     1 | Cost: 4.429231e+03Iteration     2 | Cost: 8.254558e+02Iteration     3 | Cost: 3.318581e+02Iteration     4 | Cost: 2.639346e+02Iteration     5 | Cost: 2.261522e+02Iteration     6 | Cost: 2.106102e+02Iteration     7 | Cost: 1.737319e+02Iteration     8 | Cost: 1.639049e+02Iteration     9 | Cost: 1.591224e+02Iteration    10 | Cost: 1.547121e+02Iteration    11 | Cost: 1.496355e+02Iteration    12 | Cost: 1.461199e+02Iteration    13 | Cost: 1.441541e+02Iteration    14 | Cost: 1.415434e+02Iteration    15 | Cost: 1.410112e+02Iteration    16 | Cost: 1.406926e+02Iteration    17 | Cost: 1.402903e+02Iteration    18 | Cost: 1.398782e+02Iteration    19 | Cost: 1.394818e+02Iteration    20 | Cost: 1.390831e+02Iteration    21 | Cost: 1.386609e+02Iteration    22 | Cost: 1.384726e+02Iteration    23 | Cost: 1.384556e+02Iteration    24 | Cost: 1.381757e+02Iteration    25 | Cost: 1.380535e+02Iteration    26 | Cost: 1.379410e+02Iteration    27 | Cost: 1.379023e+02Iteration    28 | Cost: 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| Cost: 1.374669e+02Iteration    57 | Cost: 1.374667e+02Iteration    58 | Cost: 1.374663e+02Iteration    59 | Cost: 1.374659e+02Iteration    60 | Cost: 1.374656e+02Iteration    61 | Cost: 1.374651e+02Iteration    62 | Cost: 1.374648e+02Iteration    63 | Cost: 1.374647e+02Iteration    64 | Cost: 1.374645e+02Iteration    65 | Cost: 1.374643e+02Iteration    66 | Cost: 1.374642e+02Iteration    67 | Cost: 1.374641e+02Iteration    68 | Cost: 1.374641e+02Iteration    69 | Cost: 1.374640e+02Iteration    70 | Cost: 1.374640e+02Iteration    71 | Cost: 1.374639e+02Iteration    72 | Cost: 1.374638e+02Iteration    73 | Cost: 1.374637e+02Iteration    74 | Cost: 1.374637e+02Iteration    75 | Cost: 1.374636e+02Iteration    76 | Cost: 1.374636e+02Iteration    77 | Cost: 1.374635e+02Iteration    78 | Cost: 1.374635e+02Iteration    79 | Cost: 1.374635e+02Iteration    80 | Cost: 1.374635e+02Iteration    81 | Cost: 1.374634e+02Iteration    82 | Cost: 1.374634e+02Iteration    83 | Cost: 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1.374632e+02Iteration   194 | Cost: 1.374632e+02

Training accuracy of our algorithm using linear regression

fprintf('lambda = 2.5\n');
% Applying learnt parameters on test data
pricetr = Xtrain * theta3;

% Showing algorithm accuracy
errortr = abs(pricetr - ytrain) ./ ytrain;
accuracytr = 100 - (mean(errortr) * 100);
fprintf('Training accuracy on training set: %f\n', accuracytr);

lambda = 2.5
Training accuracy on training set: 87.823106

Cross Validation Accuracy of our algorithm

%disp(size(Xtest));
pricecv = Xval * theta3;

% Showing algorithm accuracy
errorcv = abs(pricecv - yval) ./ yval;
accuracycv = 100 - (mean(errorcv) * 100);
fprintf('Cross Validation accuracy on cross validation set: %f\n', accuracycv);

Cross Validation accuracy on cross validation set: 88.760413

Testing our learnt algorithm using linear regression

% Applying learnt parameters on test data
%disp(size(Xtest));
pricete = Xtest * theta3;

% Showing algorithm accuracy
errorte = abs(pricete - ytest) ./ ytest;
accuracyte = 100 - (mean(errorte) * 100);
fprintf('Testing accuracy on test set: %f\n\n', accuracyte);

Testing accuracy on test set: 71.199640

Polynomial Regression with Regularization

Contents

Clearing and closing the figures

Loading data set

Feature Mapping for Polynomial Regression

Creating training and testing datasets for the learning algorithm

Learning Curves (lambda = 0)

Learning Curves (lambda = 1)

Learning Curves (lambda = 2)

Learning Curves (lambda = 3)

Validation Curve

Training Linear Regression with Regularization (lambda = 0)

Training accuracy of our algorithm using linear regression

Cross Validation Accuracy of our algorithm

Testing our learnt algorithm using linear regression

Training Linear Regression with Regularization (lambda = 1)

Training accuracy of our algorithm using linear regression

Cross Validation Accuracy of our algorithm

Testing our learnt algorithm using linear regression

Training Linear Regression with Regularization (lambda = 2)

Training accuracy of our algorithm using linear regression

Cross Validation Accuracy of our algorithm

Testing our learnt algorithm using linear regression

Training Linear Regression with Regularization (lambda = 2.5)

Training accuracy of our algorithm using linear regression

Cross Validation Accuracy of our algorithm

Testing our learnt algorithm using linear regression